A Treatise on Spherical Astronomy - Forgotten Books [PDF]

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P RE F A C E herical A stro o my I m ea that par t f M t h m t i l s ro o my w hi ch lies b e t w ee t h t d m ai f D y a m ical A stro o y o d the m ul t itu d i ous d etails f t h e o e ha d P ractical A s t ro o m y o t he o t her I have ai m ed at provi di g f the stu d e t a book Spherical t he li m its thus i dicate d A s t ro o my w hich is ge erally w i t hi but I have t hesi t a t e d to tra sgress t hose lim its w d t he w he there see m e d to be goo d reaso f d oi g so For e am ple I have j ust crossed t he border f D y a mical A stro om y i d i t w o co cludi g cha pt ers I have so f C ha pt er V I I e t ere d P rac t ical A s t ro o m y as t give som e accou n t o f the fu da m e t l geo m etrical pri ci ples f astro m ical i s t ru m e ts I t has b ee assu m e d t hat t h rea der f this book is already acquai te d w ith t he m ai fac ts o f D escri ptive A s t ro o m y Th read er is also e pecte d t o be fa miliar w i t h t h ordi ary processes f P la e d S ph erical T rigo o m etry d he shoul d h ve at leas t ele m e tary k w le dge f A aly t ic G eo m etry n d C o ic S ectio s as well as f t he D iffere t ial d I t egral C alculus I t n ee d har dly be added t ha t t he stu de t f y bra ch o f M a t he m ati c al A stro my shoul d also k w the pri ci ples o f S t atics d D y a m ics A a guide to the stu d e t w h o is m aki g his first acquai ta ce w ith S pherical A stro omy I have a ffi ed a asterisk to t he t i t le f t hose articles w hich he m y o m i t o a first readi g ; the ar t icles so i d ica t ed bei g ra t her m ore adva ced tha t he articles w hich rece d e or f ollo w p S uch ar t icles as relate to the m ore i m por t a t subj ec t s are ge erally illustrated by e ercises I n m aki g a selectio fro m the large am ou t o f available m aterial I have e deavoure d t o choose e ercises which o t ly be a r directly t he te t but also ha e so m e special stro o m ical or m a t he m a t ical i terest It will be see t h a t t he T ri pos e a m in ation s at C a mbri dge d m a y C ollege e a m i atio s at C a mbridge n d elsewhere have provided a large f d f m ro ortio the e ercises I have also ob t ai e e ercises ro x p p m a y ther sources w hich are duly i dica t ed Y Sp A t n m ’

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the subj ect to w hich I have most freque tly m B i i w t ur e re ari g this volu e is s h S l e r i c p p p E glish A t o my a m os t e xcelle t book w hich is available i i t s origi al Ger ma d Fre ch t ra slatio s as w ell as i e xte sive A m o g rece t au t hors I have co sul t e d V l t i e stu de t f as t ro o m y H dw te bu c h d er A s t n o mi e which a fford t overlook d I have lear ed m uch fro m the admirable wri t i g o f P r fessor N ewcom b I have t ack o w ledge w i t h m a y t ha ks t he as is t a ce which M A rthur B erry has f frie d s have ki dly re dere d t m e i h e d m e w i t h m a y solu t io s f e ercises m ore es pecially o f D J L E D reyer has rea d over t he chapt er T ri po questio s M W E H artley d m a d e use ful sugges t io A berratio has hel pe d i the c rrec t io f the proo fs as w ell as i t h e revisio f par t s o f the m a uscri pt M A R H i ks has gi e m hel p i correc t io f t h e proo fs d I m also i deb t ed t o hi m f th assis t a ce i t he c h apt er o the S olar P aralla D A A R a mb t has devote d m uch ti m e t o t he rea d i g f proo fs d has assisted i m a y other ways M F J M S tra t t o h revised so m e f t he pages especially those the rotatio f t he m oon D E T W hit t aker has give m e use ful sugges t io s es pecially i t he chapt er R e frac t io d he has al o hel ped i rea di g proo fs d my o M R S B all has d ra w m a y o f t he d iagra m s L astly I m u t ack owle dge my obliga t io to the S y d ics f the U iversity P ress w h have m e t all my w ishes i t he ki d es t m a er Th list f paralla e m ore e t e f stars ( p 3 2 8 ) is based sive lis t s give by N ewco m b i The St s d K pt y i the G ro i ge publica t io s N o 8 Th e resul t s state d f C e tauri S irius d a G ruis have bee n ob t ai e d by S i D G ill ; t ho e f P rocyo A l t air A l d ebara C a pella V ega A rcturus by D E lki t hat f C or doba Z o e 5 2 4 3 by D D e S i t ter ; t ha t f 1 8 30 G roo mbri dge by P ro fes or K pt e y ; t ha t f 2 1 1 8 5 L ala de by M H N R ussell ; t ha t f d t ha t fo P olaris by P ri t char d ; 61 C ygn i is a m ea resul t I ought to dd t ha t whe I use the word e phe m i s I re fer so f as works i the E glish la guage are c cer e d either to the B i ti sh N ti c l A l ma Ephe me i s c or to t he A me i c Th e n d

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F UN D A M E N TA L F O R M ULA E 1 b h er ic al T r i m t r y 2 D ela m re d N a i er a al i e l S g p p g § lvi g pherical t ia gle 3 A ccu racy atta i ab le i L gar ith m ic f 4 D i ffere ti al fo r m u lae i a S pher ical T ri a gle 5 Th C al cu lati E xer ci e C h apt r I A t f I te rp lati ono

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T H E U SE O F SPH E R CA L CO O R D NA T E S

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f ap i t rad uat d great ci rcle th 7 C rd i at ph re t w p i t th x r i h c i e f t h b e t wee 8 E f t ph re p § f th term f th ei r c rd i ate 9 I t erpre ta ti ph ere i equati i ph er ical c rd i ate 10 T h i cl i ati f t w grad uated 11 le O th i t g r at ci r c le i t h t 1 80 j i i g th i r ti 12 T ra f r mati f c rd i ate f t w grad u at d gr at ci r cle 13 Adapta ti t l gar ith m

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43 F I GURE O F T H E EA RT H A ND M A P M A K IN G 14 I t r duct ry h a titu e a i f cu rva tu re al t 1 5 L d 1 6 R d g § m er id i a th at a m p 18 C d iti g 1 7 T h th e ry f m p m aki g 20 hall b c f rm al cal i a c f rm al repr e tati 19 T h Merca t r pr j ecti 2 1 Th e l x dr me 2 2 S t r g raphic p 2 3 Th t re graphic pr j cti f th j ti ph re i y ci r cl al a ci rcle 2 4 Ge eral f rmulae f t re graphic pr j ecti 25 M p i w hich ach b ar a c ta t rati t t h c rre p di g area t h E xer ci e C h apt er I II ph ere

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C ELE S TIA L 82

3 2 Th fi r t p i t f A r i e i ht a c i d d ecl i a ti § 4 D e t r i a ti d 3 m f e ith h d d t h u r a le i er al T h 33 y 5 g § d dec l i ati 35 A ppl icati d a i muth fr m h ur a gle d i ta ce f a c le ti al 36 O t h ti m e f cu l m i ati f t h d i ff r ti al f r m l ae 38 C le ti al latitu d d etti g f a c ele ti al b d y 37 R i i g b dy C h a pt r V d l g itude E xerci

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M OS PH ERI C RE F RA C T I ON 11 6 m ical r fra cti law f ptical refracti 40 A tr 3 9 Th f m f 4 2 eral th r y a t h er ic re ra cti I t ra ti th e f 4 G 1 p g § t h refracti 43 C a i i f r m u la f at m f d i ffere ti al equati 44 O th er f r m lae f atm pheric r fracti ph er ic refr cti E ff ect f atm ph eric pre u re d t mperat re r fracti 45 46 O th de t rm i ati f a t m ph eric refracti fr m b ervati h u r a gle d decl i ati 4 8 Effect f § 4 7 E ff ct f refracti refracti th appare t d i ta ce f t w ei ghb u ri g cele ti al p i t 4 9 E ffect f refracti t h p iti a gl f a d ub l tar M i ella e u refracti que ti AT

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l u i lar prece i i b erv d 5 5 Phy ical ex pla ati f lu i lar prece i utati d 56 P la etar y pre ce i 5 7 G eral f rm ulae f d R i g ht A c e i d uta ti i pr ce i D cl i ati 58 M veme t f t h fi r t p i t f A i th E c li ptic 59 Th i dep d t d y umb r f S tar 60 P p r m ti 61 V ar i ati i t rr trial latitude E xerci e C h apt r V III 54

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P A GE

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M EA N T IM E tti g f t h a tr 65 E ti mati f

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63 Th i ereal ti m mical cl ck b l i qui ty f t h e l i ptic 64 T h a curacy th b tai ab le i th d ter mi ati f R i g ht A c i i dereal 66 Th 67 Th year d t h t r pical year e e t r ic al r i ci le a ea m f m p p g m ti 68 69 Me ti me Th idereal ti me a t m a f m ea ti m e fr m i dereal ti me 7 0 D e term i ati t erre t r i al 1 T h 7 e s g da te l i e E xerci es n C h apte r I X 62

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M O TI O N 2 26 t t h equa t r 7 3 T h eq u ati 72 Th r d ucti f t h ce tre f ti m e ected with t h equati 75 F rm u l e c 7 4 T h equati G raphical r pre e ta ti f ti m f th equati 76 f ti me f ta ti ary eq uati f ti me 7 7 G eral i ve ti ga ti 7 8 Th cau f t h e E xer ci e Chapt r X T H E S U N S A PPARE N T A N N UA L ’

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8 1 A ppl ica ti elati v V l city t th c rdi at f a cele tial A b errati f A b erra ti 8 2 E ff ct 84 Ab i er t ki d f A b erra ti rrati i R ight b dy 8 3 D ff e g i L g itu d d L atitu de A ce i 8 5 A b errati d D ec l i ati f th 8 6 Th 87 E ffect E ll i ptic G e me t ry f a u al A b rrati f th c M ti f t h E rth A b rrati ta t f 8 8 D eterm i ati A b rrati 90 P A b la e t ar y rrati 89 D iu r al A b rrati g F rmu lae f red ucti fr m ma t appare t place f S t r 91 E xer ci C hapter XI

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I NSTRU M E NT 431 140 Fu dam e tal pri ci ple f t h ge eral i ed i t ru m e t g 14 1 Th l i e i t h ge eral i ed i trum e t repre ted p i t th ph ere x re i h c r d i a t e f a c l ti al b d y i t r m h 1 4 2 E f t f t p § r ad i g f t h ge eral i ed i trume t whe d i rect d u p it 143 I ver e f t h ge eral i d i t ru m e t 14 4 C f rm f t h fu dame tal equati tra t b twe t h d i rect d t h i ver pr b lem f t h ge eral i d D et rm i ati f t h i dex err r f ci r cle II i i t r u me t 145 th 14 6 Th de t rm i ati f q d by b rvati ge eral i d i t r u m e t d left p iti f t h i trume t f tar i b th r i g ht 14 7 D e ter 14 8 D eterm i ati m i ati f A d 6 f t h i d ex err r f ci rc le I 14 9 O a i g le equati which c mpr i e t h the ry f t h fu dame tal 1 5 0 D i ffer ti al f rm ulae i t h th ry i t rum e t f t h b rvat ry f th 15 1 A ppl icati f th d i ffere ti al ge eral i ed i t ru m e t f r m ul e 15 2 Th ge eral i ed t ra it ci rcle T H E G EN E RA L Z E D

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n

f To d

oo

.

a

n

n

,

s

.

o

s n

'

hu t n

.

’

er s

Sp h e r i c a l T r i g o n o me tr y

p 27 .

1— 2

.

F UN D A M E N TAL F O RMUL A E

[

OH

.

I

t here is s till n o t hi n g to sho w w hich o f two su pple m e n tary values is t o be give n to C a n d u n less so m e a dditio n al in for m ati o n is ob t ain able showin g w hether 0 is acu t e or obtuse t he proble m is am biguous I f t w o a n g les a n d a side o ppo site to o n e o f the m are give n then fro m for m ula ( 3) t he si d e opposi t e t he o t her an gle w ill be d eter m i n e d s ubject a s be fore to a n a m bigui t y bet w ee n t he arc If

si n

'

O< 1

,

,

,

.

,

,

,

its su pple m e t Whe t he am biguity i ei t her case is re m oved the pr ble m re d uced to t hat i w hich tw o si des d the a gles o pposite t o both are k o w Fro m equatio ( 1 ) d ( 2 ) t he follo w i g for mula is easily de duced t a cos 0 + t 0 cos A an

d

n

.

n

n

o

an

n

n

n

n s

.

n

an

n

an

an

— ta n a co s

l

C ta n

A

c co s

’

will show w hether b or l 8 o + b is to be calculation m y be si m plifie d by taki g

an

°

( 2)

d

n

a

ta n 0 = ta n

whe ce we fi B y m ea s

n

n

n

d b o

a co s

C ; tan ¢ = tan

A,

0

the polar t ria gle we obtai t A cos c t C

f

c co s

n

n

an

an

co s a

rom which B m y be deter m i e d f ( 5 ) re m oves the a m biguity bet w ee B a d 1 8 0 + B A lso i f we take A cos c d t t t 9 t 0 cos a we fi d B 1 80 0 gb Whe t he three si d es are give a s pherical t ia gle m y be olve d as follows L e t 2 b c the f

a

n

°

n

n

an

or

.

’

an

an

an

'

°

n

an

,

’

.

n

n

s

s

.

tan

a

A 5

r

,

n

n

,

s i n 3 si n

(3

a

)

by which A is fou d d by si m ilar for m ulae we ob t i B I f t he three a gles A B 0 w ere give t he m aki g n

n

w e have

by which

ta n

a

is fou

n

a n

an

,

,

n

,

n

,

%a d,

an

d

si m ilarly f

or

a

b

an

d

c

.

n

an

d C

.

F UN D A M E N TA L F O RMU L AE T h e rela t io here i m plie d be t wee t h e right a gled t ria gle d h w by Fig 4 I f A B d t he qua d ra tal t ria gle i N apier s t he 4 C B C i pro d uce d t C so t ha t B 0 rules applie d t t h right a gle d t ria gle A C C give t h for m ulae belo gi g t t he quad ra t al t ria gle n

n

s s

n

n

an

o

s

n

o

.

”

'

’

-

n

n

an

.

’

'

n

'

e

o

,

n

o

n

n

n

n

-

e

,

n

FIG

4

.

.

RIT HM S — Th u ual ot t io e m ployed i writi g log i t h m s f the trigo o m etrical fu ctio s m y be illustrate d by e am ple Th a t ural cosi e f 2 5 is 0 90 630 78 d log cos 2 5 log log 10 0 0 42 7 2 4 To obvia t e the i co ve ie ce f egative l ogarith m s this is so m e t i m es writ t e w hich s t a d s f LO G A

ar

a

n

x

n

n

n

n

o

an

s

e

:

n

n

a

.

e n

°

o

n

an

°

.

n

n

n

n

o

n

n

n

or

0 9 5 72 76

1

.

We shal l however ge erally f ll w t he m ore usual practice f t he table d dd 1 0 to the logarith m f every trigo o m etrical fu ctio Whe this ha ge is m ade w e use L i stead f l i w i t i g t h e w ord l g T hus i the prece di g case the L g w oul d be w ri tt e 99 5 7 2 7 6 d m ore ge erally L g cos 9 log cos 9 10 I f i t is ece sary t s t a t e t ha t the t rigo o m e t rical fu c t io f which t he log arith m is used is a egative u m ber i t is usual to wri t e ( ) a ft er the logari t hm F e ample i f 1 5 5 occurre d as a fac t or i e pres i 2 7 6 ( ) as its L o g arith m w here t h figu es w e shoul d wri t e de ote L g cos n

s

n

n

an

c

o

n

n

n

.

o

.

o

n

n

n

n

o

n

.

x

,

co s

°

n

n

n

n

n

n

or

o

n

o

s

n

n

an

n

o

o

n

r

o

a

n

.

o

o

,

an

x

e

s on

r

F U ND A M EN TA L F O RMU L AE

I t freque t ly ha ppe s t ha t a ft er a gle 9 has bee n d e t er m i ed i the firs t par t f a co m pu t ati we have t e m ploy cer t ai trigo o m e t rical fu t i s f 9 i t h seco d part f t h sam e co mpu t a tio I t hi se o d par t we have fte a choice as to whe t her we shall e mpl y o for m ula de pe di g L g si e L g or a o t her d e pe di g I t is ge erally i m m aterial s 9 whic h for mula t he calcula t or e m ploys bu t i f 0 be early ero or early 90 n o f t he for m u l ae w ill be u cer t ai d the other shoul d be use d It is there f re proper t o co si der t he which t he choice shoul d be e ercise d i so f as pri ci ples d ge eral ri ci les be lai dow n y p p W e m y assu m e t hat pr per care havi g bee t ake the w ork is free fro m u m erical err r so f as the eces ary li mita t io s f the tables w ill per m i t B t the e very li mi t a t io s i mply that the value f 9 w e have ob t ai e d is o ly appro i m ate value Th e c lculator m a y ge erally pro t ect t h latter par t o f t h w ork fro m beco m i g appreciably wro g otwiths t a di g that it is base d a qua ti ty w hich is o m e w hat erro eous Th e practical rule t T h e t wo q a n t ities L o g i 9 follo w is a very si mple e d d the for m ula co t i i g the t ge e lly equal L g cos 0 are grea t er should be use d i t he re m ai der f t he calcula t io tha t i f 9 be T his follows fro m t he c sidera t io 45 a s m all error i (9 w ill have le s e ffect i 0 ( cos 9) tha cos 6 ( i n

n

n

an

n

n

o

n

n

n

nc

on

s

c

n

.

on

o

n

n

n

e

n

n

o

ne

on

n

o

o

n

co

o

o

n

n

n

o

n

e

n

ca n

a

n

o

,

n

ar

a

n

,

x

.

on

o

.

u

.

ra

s n

an

n

n

n

on

a n

.

S

.

h wh o

n

t he

s

i de

on

c o t a si n

i f we

a re

g

ma y b e d

a

b=

tA

oo

si n

s n

n

etermi

ne

r ula

d b y t h e fo m

0 + co s b

co s

0

i ve

n

A = 1l 7

C = 15 4

°

' = b 1 0 8 30 30

°

°

9 6 38 2 2 30, 9 34 88 4 7 4

(n ), A

sin

0

co s

0

co s

b

t

a

si n

b

Lo g c o t

a

si n

b

Lo g

si n

b

(N a t ) c o t .

co

Lo g

co t a

.

°

s

ow

n

o

s n

Ex 1

an

n

n

o

n

n

n

n

n

e

n

on

,

s

an

s

n o

n

e

n

o

n

n

,

n

n

n

n

n

o

ar

.

s

u

.

n

n

o

an

n

x

n

z

n

o

on

an

n

n

.

.

n

o

on

,

°

e

a

= 8 6 13’ °

on

F U N D A M EN T AL F O RMUL AE Ex 2

Be

.

.

a n d B b y th e

D

raw GP

= 5 7 4 2 39 19 b i ve g g meth d f r ig ht a g l d t ri a gl A B t r ; th e ) p p p

i

°

n

n

o

o

e

-

9 9 2 70 4 3 2

A

9 9 3660 7 7

si n

0

°

18

:

es

n

e

n

°

’

12 0 12

'

.

n

o

.

Lo g si n b si n

'

= 46 5 5 ’ 5 8 p °

p

t a n b 10 19934 5 4 cos

—A 1 8 0 ) (

cos

tan m

9 9 0 10 608

co s

p

9 8 34 32 91

(c + m)

9 7 2 62 68 4

m

a

co s a

c o ec ( s

c

°

’

3 8 31 45

68 40 4 8

+ m)

D e l amb r e

’

s

an

d N a pi e r

’

f

n

of

n

n

l

an a o

s

ollo w i g equatio s are a stro o my Th e

FIG 5

’ = B 5 1 38 5 5 °

t a n B 10 10 1 7 105

.

’

a t. p 10 0 2 932 18

-

2

°

gi e s

.

g eat utili ty

s pherical

in

r

°

si n

g

e

cos “ A

si n

co s

— b)

flA

c si n

B)

$0

si n

4(a

si n

b) —b )

} (A

c si n

g

1

hese equatio s are o ft en described as G auss a alogies but t heir discovery is really d e to D elam bre A D ela m bre s a alogies are m ore co ve ie t fo logarith m i f calculatio tha they are o te d 3 ( ) f d f re er e t he solutio f pherical tria gles whe n a b d C p are give or w he A B d c are give I t is freque tly t roubleso m e to re m e m ber these for mulae without such assista ce as is give by R mb t s u l e l W e w ri t e the t w o rows f qua t ities T

’

n

n

,

u

’

s

n

r

n

n

n

n

n

c

r

n

an

or

n

n

n

s

o

n

an

,

n

an

,

’

.

n

’

n

n

o

a

Fo r

thi t t m t s s a e

en

b e ma de t o M r L e a t h e m

as ’

b)

.

we

s ed

ll

iti

as on

,

we

Ma

b)

,

a

r

fo r t h e p o f To d

o o fs o f

hu t r n

e

’

s

th

e se

r ula r f r c

fo m

e,

Se e D r A A Ra mb a u t , A s tr o n o mi s c h e N a c hr i c h t e n , N o 41 35 .

e e

en

Sp h e r i c a l T r i g o n o me tr y

.

.

.

.

p 36 .

'

n

—B i ) sc

w

r

au

'

.

.

e

ma y

F U ND AME NTA L F O RMUL A E

1 —2 ]

where

C = 1 80 ’

(

T

he

i r i n d e n c e e ) ( fi

Ra m b a u t

n

'

Su m c o si n e

°

si n e

o the r

th e

) in

is

r ow

on e

r ow

’

a

rule is as follo w s

s

lw a ys t o be

a sso ci a t e d

w i th

.

For e xam ple to obta i t he D elam bre a alogy w hich co tai s t rule i 4( A — B ) we co clu d e fro m R m b d B e ter ( 1 ) that 40 m ust e n t er wi t h a si n e because A as a diffe e ce ; that n d b m ust e ter as a dif e e ce because 4 ( A B ) 2) e ters w i t h a si n e ; — b m us t e ter w ith a si e because A a d B that a 3 ) ( ) 4( e ter as a diffe e ce ; ust e ter with a si e because d e ter m n 6 4 that 0 4 ( ) as a diffe e c e H e ce the a alogy ma y be writte d ow i 4c i 4 ( A — B ) = si 4C i 4 ( — b) cos 4C s i 4 ( b) e xa mple f the use f D ela mbre s an alogies we ma y A e m ploy the s pherical tria gle i w hi h n

n

n

s n

a

n

n

’

au

s

:

an

n

r n

a

n

a

r

n

n

n

r

n

n

n

n

a

a

n

a

°

62 4 8

c

'

6 = 5 7 4 2 39

,

'

°

A

93 4 6 36

B

2 9 30

.

,

We shall su ppose t ha t b C are give u m erical values here set d ow Th correspo di g trigo o m etrical fu c t io s ,

,

n

0 4

4(

a

si n

+ b) = 60

°

’

14

°

35

15 46 5 ; ‘

b)

8 6 4 8 62 8 6

cos 40

99 8 5 7 5 7 8

4 (a

8 634 38 64 si n

4(

a

s in

b)

9 9 38 67 5 2

0 4

9 4 0 1 330 1 93 4 00 0 5 3

cos 4 ( b) cos 40 a

n

an

n

d fin d A B ,

are t he

L

’

4(a

b)

2

°

33

’

si n

40 s i n 4( A

B)

si n

40 cos 4( A

B)

99 99 5 690 99 8 5 7 5 7 8 99 8 5 32 68

cos 4c si

n

4( A

an

ogs

n

n

n

n

,

1 1 13

46 6 a

.

’

n

n

=

s n

o

o

a

’

n

s n

an

n

n

n

n

n

s n

e

a

.

n

s

n

n

”

r

n

B)

o

d f

c

.

the

F UN D A M ENT AL F O RMUL AE

cos 4 (

si n

'

9 6 9 5 4 9 99

0 4

9 4 0 1 330 1

cos 40 cos 4 ( A

cos 4o i 4( A + B ) cos 4c c s 4 ( A + B ) s n

9 98 5 32 68

o

9 09 68 300

4 (A

B)

0 8 8 8 49 68

4 4c c o s 4 ( A t a n 4 (A

B)

8 634 38 64

tan si n si n

*

b)

a

si n

0 si n

4

c co s co s

0 si n

4

si n

4( A

—B

9 2 94 38 1 1

—B A ) 4(

93 4 0 00 5 3

— B) A 4( si n 40

9 9 91 7 3 5 2 9 3 48 2 7 0 1

4(A + B )

99 8 5 32 68

4 (A

B)

99 9640 1 2

cos 4c

99 8 8 9 2 5 6

4 4c 40

9 3 48 2 7 0 1

co s

ta n

H e n ce

82

—B )

11

38

’

3

8 33

+ B) —B

3 11

)

8 33

B = 7 1 2 9 30

99 8 8 9 2 5 6

40

9 3 5 9 344 5

A = 9 S 4 6 36 °

4 (A 4(A

9 34 000 5 3

0

+ B)

°

A = 93 4 6 3 6

) B)

si n

4( A A 4(

B)

B = 71

’

’

°

2 9 30

c

12

°

'

53 3

= 2o

Fro m D ela mbre s a alogies w e ily obtai the follo w i g fou r for m ulae k o w as N a pier s a alogies ’

e

n

as

n

n

’

n

n

n

4( A

cos

A B + ) 4(

si n

4(A

4 (A cos 4 ( a cos 4( a si n 4 ( a si n 4( a si n

e xa mple f the solu t io an a logies w e m y t ake As

an

—

co s

o

n

—

B) B)

+ B)

b) b) b) b) o

f

ta n

4c

t an

4c

cot 4C cot 4C

a tria gle by n

a ier s

N p

’

a

A = 2S

W e u se > si n 4 A — B ( ) 1 We u se c o s 4 (A + B )

h th lr dy xpl i t hi t h r th t

hi

B =7

°

as

.

s

rat

a

ea

s

ra

er

e

e

an

si n

a n ed on an

co s

°

c

—B A 4( ),

c si n

4

p

4c

.

7

= 74

°

be

cu

be

cu

a

se

co s

4 (A

si n

4 (A + B ) i s

-

B)

is

.

a

se

F UN D A M E NT A L F O RMUL A E an

d

fo r

use four fig l gari t h m s w hich are quite accurate e ough m a y purpo es cos 4 ( A B ) 9 9 95 6 i 4( A B ) 9 148 9 u re

-

n

s

o

n

.

s n

4(A

se c

46 b)

ta n

4( a

ta n

98 8 09

= 60

a

4(

—b

fi di g

fo r

is

’

0

n

n

an

d

ta n

'

°

a

ar e

4(a 4( a

40 b)

98 8 0 9 9 607 0 2

’

°

b = 15

+ b)

4( 2 2 ( ) w hich

)

a

tan

9 8 92 3 58

As

4(A

c o se c

both 4 5 the pro per for m ula be w rit t e °

ma y

n

= co s 0 4(a 4

tan

cos 4 ( a sec 4(

b) = 9 967 1 '

b ) = 0 1 0 33 '

a

co t

B ) = 05 614

4( A tan

40

06 31 8

:

0

°

153

:

tt a i n a b l e i n L o ga r i th m i c C a l c u l a t i o n W he t he logarith m f a trigo o m etrical fu ctio is give n it is g e e l ly pos ible t o fi d t he a gle w i t h su fficie t accuracy B t w e o ft e m ee t w i t h c a s es i which t his state m e t cea es to be qui t e true For e xam ple su ppose we are re t ai i g o ly five figures i ou r logari t h m s a d t hat w e wa t t o fi d 9 fro m t he sta t e me t that 3

Ac cu

.

racy a

.

o

n

ra

n

s

u

n

n

n

n

n

n

n

n

.

n

s

.

n

,

n

n

n

n

n

n

n

L o g s i n 6’

99 9998

.

his tells us othi g m ore tha that 0 m us t lie so m e w her e w ill the rete tio f bet w ee 8 9 2 3 7 N d 8 9 31 m a y as seve places f deci m als alway preve t a m biguity We n ote f e a m ple that every a gle fro m 8 9 5 6 1 9 to 8 9 has as its L g i the sa m e t ab lar value i 9 9 999998 We thus see that a gles ear 90 are t well de t er m i ed fro m the L g si like m a er a gles ear er are t w ell d i determ i e d by t he L g cos B t all a gles be ac urately fou d fro m t he L g t will w be proved I f 0 receive a s m all i re m e t h or i circular m ea ure h i 1 d t he i cre m e t i L o g t d be x u i t s i the 7 t h place f deci mal we have t o fi d t he equatio bet wee h a d

T

n

°

n

n

n

”

'

or

s n

u

,

an

n n

o

o

an

n

s,

n

n

’

n

n

z

o

n o

c

n

.

n

10

.

ca n

n

so

°

.

n

n o

n c

v z

”

’

no

n

u

.

as

an

°

n

n

n

,

o

.

°

n

,

n

n

s

n

n

n

or

o

x

o

o

'

°

an

n

,

n

”

n

n

an

n

s n

s

o

n

n

n

as

.

F U ND AM E NTA L F O RMU L AE

[

1

.

ha gi g the co mm o logs i to N apieria logs by the m o dulus w e have 6 i 1 6 log log 4 t h t 4 3 0 3 0 1 0000 0 0 ( / h i 1 cot 0 4 343 l g ( 1 C

x

_

CH

n

n

n

n

s n

an

e

o

n

"

s n

,

an

,

0 4 3 4 3 10 g , ( 1

h si n 1 t a n

‘

whe ce by e pa di g t he logari t h m s x

n

x

which

n

n

4 34 30 00 s i n 1

"

be writte n

ma y

6

(t a n

very early

6) h

cot

n

,

a: s i n

h

d he ce the co cluded grea t est value o f h is value f 6 w he L g t 6 is give could ever be 1 wro g u less Lo g t 6 was it el f wro g to t he e t e t o f 0 0 000 0 4 2 l gar ith m u ed d t h c mputati E 1 S h w th at wh e 5 fig f t h la t deci m al t h err r f a gle d i exact t w ithi t w u it t ex ceed 5 ec d t mi d fr m i t t a g t c a E 2 I ve tigate t h ch a ge i t h val ue f a gl pr duc d b y t h alterati f u it i t h la t d ci mal f i t L g i d h w th a t u der all ci rcum ta ce it i m r accurat t det rmi e t h a gle b y i t t ge t th a b y i t i E 3 P r ve th at i f 6 i a m all a gle i t val u e i ec d i gi ve appr x i mat l y b y t h expre i c c 1 i 6 ( d h w th at ve if 6 b m uch 10 thi ex pre i wi ll t b err u b y much 1 E 4 I f 6 i a mall a gle ex pre ed i ec d h w that Th e

an

o

an

x

.

o

.

n

o

n

ne

x

o

n

s

n

s s ne

x

.

o

s

e as

.

s are

on

e

o

e

s

o

e

s

e

n

an

o

.

n

s

an

,

n

o

n

an

s

n

e

e

e

o

e

n

s n

o

on

o

e

o

an

e

o

e

an

e

s

s

s

s

n

n

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F U N D A M E N T A L FO RMUL A E

three o f t he di ff ere tials be ero t he the o t her th ree w ill also i ge eral va ish T his is evi de t fro m the equatio s as it is a lso fr m t he co sid eratio t h a t i f t hree f t he par t s o f a m d herical tria gle re ai u al t ere t he ge erally the o t her p m d ar t s ust also re ai u altere m p illus tra t io f e ceptio t o t his state m e t let G A Th seco d equa t io f ( i ) w ill i d Ab = 0 A c = o A B = O t require t ha t A = 0 th i s c s e a pher ical t ria gle u derg a m all U d er w h at c d iti E 1 ch a ge uch that A = 0 A b = o A A = 0 A B = 0 whi le b th A d A C t r 2 Fr m ( ii ) w that = 90 b = 90 whe c e A = 9O B = 90 t 2 If a pher ical t ri a gle re ce i v a mall ch a ge which d e E al ter t h m f i t th r e a gle h w that t h al terati i t h le gth f ide m u t ati fy t h c d iti th If

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wh ere t e r po l a t i o n I t he cal cula t io s o f as t ro o my use is m ad e n t o ly o f log rith m ic tables bu t als f m a y other tables such f e a mple as t ho e which are fou d i every e phem eris Th e t of i t erpola tio is co cer ed with t he ge eral pri ciples o which such tables are t o be u t ilised L e t y be a qua ti t y t he m ag i t u d e f which d e pe d s u po t he mag itude f a o t her qua t i ty W e t he say that y i a fu ctio f d w e e press t he rela t io t hus *

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F U N D AM ENTAL F O RMUL AE

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value

o ft e calle d the g men t adva ces by equal ste ps h a d each corres po d i ng value f y o fte calle d t he i m n c t o is calcula t ed with as uch accur cy as is de ma d ed by f t he pur pose to w hich t he table is t o be a pplie d The

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F U N D A M EN TAL F O RMU L AE

or m ula at A A

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e e beri g t hat t he arc f the curve is i disti guishable fro m that o f its o culati g circle i n t he vici ity f the poi t f co tact w e ob t ai the pri ciple o w hich i terpolatio is based d w hich m y be thus e pressed I f a ta ge t TL be d raw touchi g a curve at T d LM be or di ate co tiguous to T the the i tercept L M that ordi ate be tw ee the ta ge t a d the curve is propor t ion al to t he square f TL d I Fig 8 0 is t he origi is the ordi n ate o f P that y y f T the as w e have sho w P B varies as B T d t here fore as CT a lso CB varies as C T he ce if be t he absciss f P

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F U ND A M EN TAL F O RMU LA E

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F U N D A M ENTA L F O RMU L A E

two ti m e its the fu n ctio i creases fro m y to y he ce i t s a ve g e rate o f i crease pe t i m e u i t is 4 (y — y ) d as the ate i crea es u i for m ly i t will at t ai i t s w he l e g hal f the t i m e has elap e d i e whe t he fu c t io has t he value y H e ce w e d e d uce the followi g result T h e rate at w hich the fu ctio is cha gi g pe it o f ti m e at y e poch t i s hal f t he di ff ere ce be t wee the values f t he u i t f ti me a fter t d a t o u it f ti m e be fore t fu ctio a t is o fte m ad e i the Ephe m eris f a m ore ra pi d P rovisio dd a m rocess i ter olatio by givi g a itio al col i icati g f d o p p t he ra t e o f varia t io o f t he fu ctio a t the c rres po d i g m o m e t We shall illu t ra t e t his by fi d i g t he S ou t h D ecli a t io f t he oo M oo at 15 + t hours a fter G ree wich m ea S e pt 6 In

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n

’

n

1 4 1 3 15

his is a quadratic fo t d by eglectin g the las t ter m the root w e eek is fou n d t o be a ppro i m a t ely 08 6 S ubstituti g t his value i t i the origi al equa t io it beco m es

T

r

,

an

n

s

x

n

2

n

n

1 18 8

.

n

l 4 1 3t ‘

n

F U N D A M E NTA L F O RMU L AE w he n ce t = 0 8 5 9 a n d t he required t im e is ro o t o f the quadra t ic i s irreleva n t

11

15 5 1

m

.

°

5

The

.

other

.

It is easy t ge erali e the fu dam e ta l form ula tio give above L t us assu m e n

n

of

n

n

z

n

o

i t erpola n

.

e

1) ( t

1 ) + A 3 t (t

2)

— A 1) (t t t + , (

2) (t

where A A A A A are u d e t er m i ed coe fficie t s to be so adj us t e d tha t w he t beco m es i succession 0 1 2 3 4 the y assu mes the values y y y y y res pec t ively H e ce by subs t i t utio w e have ]

0,

2

,

n

n

,

3,

,

n

n

o,

2

1,

,

,

,

,

,

1 9

A

y

A ()

A

A0

2A I

0

l ,

y

2

= A , O y

ro m w hich

f

6A

1 2A 2

4A l

2 4A

2 4A ,

yo ,

AI

91

A2

2 ( 92

90)

6

361

yo

;

“

4 61 + 1 90)

468

1

24

this mea s we ob t ai t he ge eral for m ula n

f tAI i

-

-

-

t ( t — 1 ) (t — 2 )

— t 1) t(

'

1

2

3

)

1

.

3 4

,

A3 , A4

n

n

a

will usually be very s m all y

= z

If

.

60

1

.

c

n

461

49 3

A

.

are the successive di ffere ces G e erally t he l a st ter m m y be egle ted as A2

n

A

— 2 — 3 t t ( )( )

2

in t er polatio

of

n

n

t (t — l

A

n

,

2A 2 a

A0

A4

w here

,

,

A3

yo

,

.

+ 3A 1 + 6A 2

y

,

,

n

n

By

n

1 90

we m ake it equal to ero w e have z

+ 93)

94 )

6(31

0

qua ti t ies give i t he tables are o f course ge erally erro eous t o n e t e t which m y am ou t t o al most hal f a digi t Th e n

n

a

n

x

n

n

n

a

n

F UN D A M E N T A L FO RMU LA E

a di g we ob t ai

E xp

n

n

n

8 t ( y

4 8g

,

3y2 ( 8 t

3ys ( 8 t + y an

c n seque t ly

d

n

o

y

+

a

9)

4 t — 18 t

9)

2

12t

3

2t

2

M+

e4

mb

2 49 )

(54 1)

( 2t

c

8

w e have

3) ( 2 0

( 129

3 6)

2 15) (d

3

( 5 4t =

18 t

2

,

( 8t

48 3 ;

(8t

3)

a

4t

3

3

,

2t

12t

3

a

2a )

,

2t ( d t

+

24

f by w hich we see that the value y fi the fu n ctio f argu m e t hal fw ay bet w ee t w o co secutive ar gu m e ts is equal to the m ea f t he t w o adjacen t values less f the m ea o f the two seco d d i ffere n ces o t he sa m e o e eighth hori o t al li es as the e values As illustra t io f t his m ethod we m y t ke t he follo w i g m h M d roble T e oo n s m ea lo gi t u e at ree w ich m ea oo G n p bei g give as u n der fo 1s t 2 d 3 d d 4t h M arch 1 8 99 i t is required to fi d i t s m ea lo gitude t m id ight o M arch 2 d M m If t = 0

or

n

=

a

an

n

n

n

n

n

o

e,

n

z n

n

s

an

n

o

a

n

r

n

,

n

M arch l s t

l

oon s

on

iu

g t de

2 05

n

,

n

at n oon

n

1 st D i fi

'

6 9 '

2 18 36 4 5 0 '

2 31 48

‘

1

3 1

2 4 5 14

2

'

14 4 0 6 ‘

+ 13 2 5 5 8

Th e

.

°

+ 13 1 1 18

4t h

n

2 n d D ifi

.

“

7

1 8 ‘

required result is °

’

2 1 5 ( 8 36 °

°

2 31 48

'

2 2 5 1 0 39 5

.

'

'

1 3 ( {g

11 2

n

,

ean

+ 12 5 8

3r d

n

an

r

,

°

2 md

n

n

a

n

’

a

n

n

.

18 99

n

.

’

n

o

n

o

-

n

1 4 4 0 6) '

'

.

F UN D A M E NTAL F O RMU L AE

I SE S O N C H A P I E S h w th at i 1 b y f rm u la relati g t a ph eric al t ri gle A B 0 m y b ch a ged re pe ti vel y i t 180 — b 18 0 — A 18 0 — B ec d f N apier a al gie (2 1) fr m t h 180 — C d h e c e ded uc e t h fir t * E XER C

x

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,

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three

s

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x li

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.

i

si n

al writte

ma y b e

so

fo

s

r u la

fo m

s

( 16) —b )

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c s in

flA

fi

c si n

—b

B)=

a i m ilar amb i guity

is

o

ae

e D elamb re

r

h w th at th ere rm u l d

se n s

i n t h e fo m

n

si n an

w h at

in

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o

s

f

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ig

)

remai i

i n the

n

n n

.

Ex 3 .

S

.

h w th at o

co t a

d a + c o t B d B = c o t b d b + c o t A dA ,

dB = s i n C d b

si n a

B

si n

do

co s a

b c o s C dA

si n

.

whe t a ume t h val ue t t 1 t h c rre p d i g val u h w that a f rm ula f i terp lati b a ed y y y y re pecti vel y th e e data i gi v b y t h equati Ex 4 .

a re

If

.

o,

s

s

s

2

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n

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on

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e

( 2 ) ( t l — to )

It

ufficie t t b erve th at thi i t h e i mple t expre i t w hich b v i u ly g i ve f y t h val ue y y y wh e t

is

s

n

terms f ub tituted o

s

o

o

s

o

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fo r t

s

s

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or

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1,

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(

i 1I/

ti me

i f the

Ex 6

.

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be m

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ea ured fr s

xtracti

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n

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6

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of

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ec l i ati at 6 p m Gr e wich mea ti me n

on

in the

—2

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ho w that t h e

are

u

o

t t — lz

t h e E po

n

o

n

y o) +

ll w i

e

z

1

Gr e en wi ch me a n

S

tg

.

h w th at i f t —t = t — t = l t h f rm ula o f i terpo lati o n las t example w il l r d uce t t h fu d ame tal f rm l a Ex 5

,

.

.

e

n

n

on

A pr 7 .

F UN D A MENT AL F O RMU L AE Ex 7 .

.

M

Th e

’

o on s s

em i d iam eter i -

l w

fo l o

s as

s

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1909

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h w that t h e M o

o n s se

Fr wh e V e u Ex 8 .

.

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on

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f

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1 1h 13m

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11

14

l l

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n oon on

.

of

a

on

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on

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97

o

1 1h 10m

on

.

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11

acti al part I t rp lati gi ve t h equati

m

e

R A

s

f Ve n

18 61 ‘

m dn i g

e

e

If t b e t h e fr o

4

1 909

Au

n e

16

ll wi g data fi d t h J upiter have t h ame

m t h e fo

e

oon

3

a et r

-

tr

me

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mi di m

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o

.

M

a

h

10m 4 09 2 4 + t 2 67" 00

h

13m 3 5 8 5 8 + t

11

‘

11

—05 3 1 t ( t — 1 ) '

44 7s + o 0 8 t (z s-

s-

'

I t i plai th at t mu t b ab ut 4 H e ce t h la t t rm each i de f t h equati m y b replaced b y d S lv i g t h im pl eq a ti w h ave t = 7 8 8 7 7 w he ce t h req ui r d a w r i 18 5 5 m 8 s

Ex 9 .

.

n

.

n

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s

e

e

an

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e

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a

on

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s

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e

extr t fr ac

o

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ns

e er i

m t h e Eph m

s as

fo

n

De c 2 1 .

0 hrs 12 hr s

22

.

G M T .

.

ll w o

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.

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n

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o

14

h

n

s on

2 1m 3 5 8 09 + (2 8 m °

as

n

ec

x

i

s en s on

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o

f

oon

14

7

32 04

14

35

3 8 14

15

4

16 3 1

s

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‘

‘

s

all that t h

m

,

,

.

on

s

h w fro m B e l f rm u la eglecti g d it i very t h M o at ( 18 + 1 2 x ) h u r D 2 1 190 5 w a s S

u

1 3h 39m

.

12 ’

e

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i ht A c

.

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R g

1 905

s

e s

h

s

e

s on

+ 3 8 6 (2x °'

1 ) ( 2x +

,

e R A .

C H A PT E R I I

.

I

I

T H E U SE O F S P H E R C A L CO O RD N A TE S

.

r duat d great cir cle th ph ere C rdi at f a p mt th 7 ph ere f th 8 E xpre i c i e f th b et wee t w p i t th t rm f thei r c rd i ate ph ere i I t rpretati f equati i ph er ical c rd i at 9 10 i cl i ati Th f tw th grad uat ed g rea t ci rcle i l Z } 180 j i i g th e ir 11 O t h i ter e cti f t w g rad u ated great ci rc le 12 T ra f rm ati f c r d i at 13 A dapta ti t l gar ith m G r a d u a te d gr e a t c i r c l e s o n th e s ph e r e 6 T h circu m fere ce o f a great circle is su pposed t o be divi d e d i to 360 equal pa ts by d ividi g m arks S tart i g f o m o e o f these m arks which is take as ero the succeedi g m arks i regular order w ill be ter m ed f d so o t o a t er w hich t he 3 p n e xt m k is ero so t hat this poi t m y be i di fi e e t l y t er m ed 0 or T hus w e obtai n w ha t is k ow as a g a d te d g a t d i t m y have sub rdi a t e m arks by which each i terval ci cl e o f 1 is further d ivi d ed as m y be required I s t arti g fro m ero the u m bers m y i crease i ei t her d irectio so that there are t w per fectly dis t i ct m ethods f gr a d ua t i g the a m e circle fr m t h sa m e ero m ark walki g t he outsi de f t he s phere al g a graduated A m great circle i t he d irectio i which t h e u m bers i crease t e fro m 0 to 1 will have his le ft ha d t ha t o t fro m 0 t o p le f the great circle which m y be d isti guished by the wor d o le l nd his right t hat pole f the g reat circle w hich m y be dis t i gui hed by the w or d ti o l e m ci nt w rd l b i g b l t i i t rigi l fh d k r Th Th r b i g v i l b l f t h purp w p p d ch ic f i u p l l i g th t i pr f r d wh ich m t i mm di at ly ugg t hp l G

6

.

a

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oo

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.

on

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n

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s

n

s

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s

o

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e

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n o es

o

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n

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on

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on s

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an

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ns o

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n

°

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os n

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e

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n

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n

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r

n

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n

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s s

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n

s

s

TH E U SE

O F SP

H E RI CA L

CO O

R D INA TE S

[

CH

11

.

hus whe the t erres t rial equator is co si dere d as a graduated grea t ircle f lo gi t u d es eas t war d fro m G ree wich or P aris t he orth pole f the eart h is the ole f tha t circle so g a d uate d u t h pole f the earth I f o the other d i t s a t i l i t he ha d t he equator h e gradua t e d as to show lo gi t u des i creasi g ob erver m oves westwar d t he t he ole f t h e circle so s th graduate d is the s u t h p le o f the earth d t he orth p le f the ear t h is t he t i l e Whe a poi t a s phere is i dica t ed as t he ole f a grad uated f that great circle grea t circle t he t o ly is t he posi t io rou d it d e t er m i ed bu t also the d irec t io o f gra d uati I f t he give poi t t he s phere had bee i dica t ed as t he l e o f the gra d ua t ed grea t circle t he the direc t io f ti gradua t io woul d be reversed f by defi itio t he t i l e is the right ha d f a m w alki g alo g t he grea t circle i the direc t io f i creasi g gradua t io T i d icate the direc t io 0 t o 1 a gra duated circle i t is su ffi cie t t o at t a h arrow head to t he circle 0 1 2 3 as how i F i g 9 a d Fig 10 d it w ill be e i e t to f eak the directio o f i cre si g o p Fig 9 gradua t io as t h p o si ti e directio d the dire ct io o f di min ishi g gr dua t io as the n e g ti ve directio T

n

n

n

or

c

n

n

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n

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7

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on

a

n

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0

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po i n

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0

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grea t circle o f the sphere graduated fro m 0 at n origi 0 bei g ch se f re fere ce we e press the posi t io o f a y poi t o t he s phere by the hel p f two ordi ates d 8 wi t h respect to t ha t gra duate d great circle Whe specific values are give to a n d 8 t he corre p di g p i t S o t he sphere is ob t ai e d i the follo w i g way 10 F We m easure fro m 0 alo g the grea t circle i the di e c tio f i cre si g grad ua t io t o a poi t P so that 0 P = A t P a great circle is d raw er e icul r to this P a n d o d n O p p arc is t o be se t o ff equal to 8 If 8 is positive the the An y

°

n

o

n

or

x

n

'

n

a

n

ca n

,

n

n

n

o

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n

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n

,

,

n

6—7 ]

TH E

US E o r

H E R I CA L

SP

R D IN ATE S

CO O

27

require d poi t S is to be t ake i t h e he m is phere which co tai s t he ole B t i f 8 is ega t ive t he t he require d poi t the t i l T hus whe S is i t he he m isphere which co tai the pl ce f a poi t t he phere is d efi i t ely 8 are give i dicat ed I t is o fte co ve ie t to speak f the he m is phere w hich co tai s t he ole as t he positive he m isphere n d tha t which co tai s the a t i o l ega t ive he m is phere th N egative values o f eed o t be co sidered f th ugh a poi t ye t i t w o l d ge er lly Q m igh t be i dicate d as 90 i f OC Q be m ore co ve ie tly i dicated by the m eas ure m e t bei g m de i the positive directio We he ce establish t he co ve tio that all values o f are to lie be t wee 0 d It is co ve ie t to restrict the values f 8 be t wee d 90 f this d is pels so m e a mbigui t y while still preservi g per fect ge erality Tw coor di a t es will i deed al w ays determ i e e f 8 oi n t but wi t hout this li m i t atio it will t f oll o w that o e p oi t w ill have o ly a si gle ossible air o f coor di ates For p p p e am ple 8 2 0 w ill i dicate a poi t n o t di ffere t fro m I f however we establish the c ve tio that 8 a 90 d e able to 8 sh l l ever lie outside the li mits 90 we m o e a ffir m that t o ly does air f co rdi a t es deter i e o p f h o oi t but that oi t ge eral air i a s but e p p p ordi at es Th e o ly e ce pt io s t heu re m ai i g will be the ole I the for m er 8 d l e f the fu da m e t al circle ti bu t i each is i d eter m i ate d i the latter 8 n

n

n

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.

n

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n

n

.

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a

.

re t ricti that 0 i i 360 d h w th at t h p i t w ul d h ave b e equally repre ted 8 re pecti vel y f val ue f y f t h f ll w i g pa i r Ex 1 .

s

o

n

ar

n

o

n

,

x

n

.

an

n

n

n

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n

on

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n

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a

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n

n

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a

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n

n

n

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,

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n

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n

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o

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a

a n

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n

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,

n

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n

n

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Ab a n d o n

.

o

o n

e

o

e

o

o

i

g the ° a = 4o ,

n

or a

s

>

e

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n

>o

on s

s

°

an

se n

n

by

s

,

alway appl y t 36o t either b th f t h c rd i ate with ut th er b y al teri g t h p iti f t h p i t t which th e e c rd i ate refer f c r d i ate E 2 S h w th at t h f ll w i g pa i r We

ca n

s

n

e

x

.

°

e

e

o

.

o

o

s o

n

360

o

s

n

oo

(1

8

a

8

°

oo

e

o

o

o n

e

on o

os

or

o

oo

n

°

18 0 + a ;

s

i d ica t t h e ame p i t d thu v ri fy th at fo every po i t a pair o f c rdi at s b e f u d uch th at O il :l 360 a d 90 e

oo

o n

s

n

e

ca n

,

s

an

o

n

s

.

18 0 — 8 —18 0 ° — 8

18 0 + a

n

s

°

°

a ll

o

s

n

n

r

e

>a

>

°

n

on °

t he

2h)

here

sp

U SE O F

THE

8

E x pr e s s i o n

.

H E RI CA L

e

c o si n e

CO O

R D IN AT ES t

f th e

o

b e ween

t e r m s o f t h e i r c o o r di n a t e s L t A A be t he grea t cir le f re fere ce le t S d S be t he t wo poi ts A A S 8 we m ust have S F = 9 O — 8 d i like We have also er S P 90 ma d PA d P A are AA e ach po i n

t

th

f

o

SP

in

s

.

'

c

e

’

n

an

o

nn

'

an

,

A

d

n

an

an

as

SP S

’

'

'

a

a

.

lyi g f da m e tal for m ula ( 1 ) t t he t ria gle SP S w e have i f S S 9 cos 9 i 8 i 8 cos 8 8 cos ( ) ( i) Whe t he poi ts S S are c l ose t ogether the sphere a m ore co ve ie t f r mula t he d eter mi ati o f their dista ce is fo fou d as follows We have cos 0 i 8 i 8 cos 8 cos 8 cos ( ) 1 i 8 si 8 {cos i ( ) 2 ( cos 8 cos 8 {cos } ( cos ( 8 cos ) cos ( 8 S ub t racti g this fro m 1 cos é ( i ) }( w e have A pp

an

°

a

a

,

s

,

’

P i t s n ole

n

.

°

'

tw o

n

u n

n

o

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n

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'

co s

,

n

o n

n

r

n

.

’

n

n

a

a

n

o

n

n

on

FIG

11

.

.

.

s n

'

s n

’

a

'

2

n

s n

'

a

'

a

a

s n

’

2

a

1

2

a

7

a

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2

'

si

)

2

n

a ( }

1

2 i n s 5 (a

a

n

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a

“

si n

2

é

9

co s

; (a a

'

a

i s ) n

’

5(8

2

n

a

1

si n

.

his is o f course ge erally t rue the a ppro xi m a t e solutio

T

s n

a

2

,

an

2

w he n

d

a ( 5

0

’

a

) cos 1, ( 8 2

1

is ve y sm all it gives r

n

9

2

We

can

L e t SN

(8

0

0

2

00 3

2

56

7

rove this for m ula geo m etrically as follo w s ( Fig d S N be per pe dicular to S P d SP res pectively is a very s m all tria gle

p

.

an

As SN S ’

:

'

’

’

’

n

an

n

SN

’2

+

N S ’

'

2

= SS

whe ce a ppro i m a t ely n

x

8 (

(a

’

a

)

2

cos

2

'

8

SS

.

US E O F SP

THE

HERICAL

COO

RD INA T ES

[OH

II

.

x lai h w t h luti f t h la t que ti appli t b th th fr m t h ti l i f t h p iti ve le h w i ti ui h le t d w h d g p c d b fr m t h fi r t tar t t h d i recti th 5 S h w th at i f L b t h l g th f t h arc f a gre t ci r c le E earth ( upp ed a phere f r d i R ) exte d i g fr m l t M l g l t lat X l g Z th e Ex 4 .

o

E p

.

s, a n

s

on

e

x

on

2,

.

L =R

tan

2

co s

co t

d)

o

e

n

A,

l

(

si n

co t

A,

‘

e

s

e

(

—l

co s

A,

cos

L e t S , S 2 b e t h e t w o po o , N the OP , P 2 t h e q

e uat r

th e

os

e

no e

o

o

es

.

u s

that t h highe t latitude r ach ed

d

an

on

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2)

where an

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s

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A, e m 1 2

by t h e g

k g si n

( Fi g

r at cir cl w ill b e

e

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(l ,

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(Z,

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.

r th

po

le

(1,

Z2 )

n

A, s i n A2

si n

S, S 2

co s

x ec d part 8 pr d uc ed co s

pr ve t h latitude S equal A S OP F r m A N OS To

on

,

,

,

s

2

high t ece ary es

ss

n

,

we

have si n

N S , s i n N S, O

c ec S S E f th 6 V er i fy that i t h expre i ch a ge i f d a ther p i t 8 th ere i d 18 0 d explai w hy thi 8 re pecti vely x

an

if

N S , s i n N S2 s i n S , N S 2

si n

an

,

N OS ,

si n

S , OP ,

o

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.

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co s

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ate s

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ti o n

.

12

.

i ta ce betwee a p i t d 8 b al tered i t 18 0 ece ary n

s

e

n

ss

n

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n

an

a

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e

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n

o

FIG

.

o

°

+a

.

s ph e r i c a l

in

8

a,

co o r

.

d 8 are give Whe the as we have sh w a poi t f which these qua t i t ies are the coor di a t es is d efi i t ely deter m i e d the s phere I f w e k ow o t hi g with regard to d 8 e ce pt that t hey sa t is fy o equatio i t o which they e n ter i co j u ctio w ith o t her qua t i t ies which are k w we ha e t su fficie t d ata t o d eter m i e t he two u k ow s A y value subs t i t u t e d i t h e equa t io will give f equatio i 8 f which i ge eral or m ore roo t s c be f u d R e peati g t he process wi t h d i ff ere t values f we obtai i d e fi i t ely u m erous series f pairs f coordi ates 8 eac h f which c rrespo ds t o a poi t t he s phere If sev e ral f t hese poi t s be co s t ructe d they will i d icate a curve t race d o n

an

a

n

o

n

,

n

on

n

.

a an

n

n

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n

n o

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n

n

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n

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n

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n

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can

a

,

,

.

n

8 — 9]

TH E U S E o r

SP

H E R I CAL

COO

R D IN ATE S

the s pherical sur fa ce Th origi al equa t io m y be described as the equatio o f that curve j u t i the sa m e way as equation a aly t ic geo m etry n d y i dica t es a pl a e curve i i W e shall first sho w that i f the coordi ates f a poi t 8 satis fy t he e qua t io n

e

.

an

n

n

n a: a

a

n

s

n

n

n

n

.

n

a,

n

o

n

here A B C are co sta ts t he locus f t he poi t great cir cle o f which t h e poles w ill have coordi ates where 180 ’

’

w

,

n

n

,

o

,

n

°

a

ll be a

wi

n

a

8

’

’

,

an

d

’

,

tan

= B /C ;

’

a

‘ ‘ 2 / A /N A H B + 0

si n

’

.

We c n m ake A posi t ive because i f ecessary t he ig s o f all the ter m s c be cha ge d A ssum e t hree e w qua tities H 8 O B = H i a cos H cos such t hat A = H si cos the by squari g d a d di g I] i VA + B 0 T aki g the u pper sig we btai fro m the firs t equa t io i 8 a posi t ive qua tity there is co f sio he ce 8 is posi t ive a d as Th e seco d d t hird equatio s give betwee 8 d 18 0 d t hus cos is fou n d without am biguity d w e d i I f ho w ever w e h d take the h a ve obtain e d e solutio egative value o f H t he i stead f 8 we shoul d have h d 8 n d the two las t o ly be sa t isfied by fro m the firs t equati 1 d 8 i s t ea of T hus t here are 0 w utti g t o solu t io s + p A d these are tw o a t ipo dal p i t s 8 nd a e d uces t o T h e origi al equatio the H { i 8 si 8 cos 8 cos 8 (a )} 0 w he n ce 8 8 m ust be 90 fro m the fi ed poin t d there fore i t s locus is a great circle i E ati fied 1 S h w th a t if t h f ll w i g qu ati a

n

,

n

an

s

n

.

'

n

n

2

n

n

a

’

n

n o

°

an

s n a

,

n

a

an

n

on

n

n

n

’

an

2

s n

n

n

’

2

.

’

’

a

n

o

n

,

'

s n

an

'

a

,

'

n

n

n

an

,

u

n

n

’

,

a

an

’

a

,

n

'

n

n

,

on

’

n

’

’

°

o

n

ca n

n

a

n

’

a

n

.

n

a

,

a

a

,

n

n

n

s n

n

o n

,

.

r

'

’

'

’

n

’

co s

°

a

x

a,

,

a

’

'

an

,

.

x

o

.

.

si n

A

l cu

the

s of

o

o

e

t h e po

i t n

a,

8

th at i f D

d

Ex 2 .

a re

c

ons

2

If

.

ta t n

e

i

s n a co s

w ill b

e

“

in g

on

e eral a n

o

.

a co

s

8 = D,

all ci rcl

which t h rad iu

e of

e

uati repre e t m re th a a p i t t h cu rre t c rd i at f a p i t a ph ere a d 8 h w that t h e eq ati a re

a,

o

e

n

u

repr e t a great ci rcle which a p le s

sm

s

is

s

32 on

eq

oo

s

n

n

s no

es o

o n

n

o

on

o n

s

n

.

a

,

b

on

ta n 8 = t a n 6 si n es n

s s

8+ C oos

1

= A2 + B J+ 0 2 the

s, s

n

8+B

cos

an

o

(a

h a s t h e po

i t n

a a

) a

2 70

'

an

d 8

90

°

b

as

T H E U SE

OF

SP

H ERI CA L

CO O

RD I N A TES

t i o n o f t w o gr a d u a t e d gr e a t c i r c l e i t h e a r c :l 1 8 0 j o i n i g th e i r n o l e s f t wo d t d great circles i ge eral Th i cli a t io i g u navoi dably am biguou f it m y be ei t her o f two su pple m e t al a gles d it is o ly whe the two circles cross at right a gles tha t t h is a mbigui t y disappears f t wo g d te d great circles ee d B t the i cli atio t be a mbiguous because w always dis t i guish t hat f t he t w o su pple m e t l a gle which is to be d ee m ed t he i cli atio o f t h t wo circle Th i c l i ti o is d efi e d to be t h e a gle :l 1 8 0 10

cli n a

Th e i n

.

°

n

>

n

e

s

.

n

n

o

or

n

an

,

ra

u n

s,

n

s

u a

e

s

n

n

a

n

n

n

.

n

n

u

n a s

o

e

ca n

ra

u a

n

on e

n

n

s

n

e

.

n

n

F IG

13

.

n

na

n o

o

n

°

n

n

F IG

.

14

.

e

n

>

.

be t wee those par t s o f the circles i w hich the arrow heads are bo t h divergi g fro m i t ersectio or co vergi g t o w d s i t ersec t i I Fig 1 3 t he two seg m e ts o f t he circles d ivergi g f o m 0 d co seque tly the a gle is to be A 0 A = are 0 A d 0 A I f however we si m ply cha ge the d irecti o f the arrow hea d 0 A wi t hou t other altera t io i the figure we have the y co ditio show i Fig 1 4 where the divergi g seg m e ts 0 A d 0A t he a gle A 0 A 1 8 0 w hich is accord i gly o w co t ai t o be take as the i cli atio o f the two graduated circles t his case If ( Fig 1 3) is the grea t circle per pe d icular bo t h to 0A the si ce 0 A = 90 we have d 0A d GA = A A If N d N be t he oles f 0 A d GA res pectively we have A N 90 d A N d he ce n

n

-

n

an

on

n

n

an

n

n

an

,

,

an

n

n

n

n

n

.

n

n

an

n

e,

2

n

n

on

,

°

n

n

.

,

n

,

e

,

-

,

n

n

n

r

,

on

n

n

n

ar

n

an

,

2

n

n

n

n

n

.

.

,

-

in

.

n

.

an

1

,

e

2

,

1

.

an

I

2

M In an

tin

like

o le

o

f

n n

n

,

2

an

an

2

A 1 A2 =

N,

a er i Fig 14 t he for m er case

m

o

n

an

an

,

2

°

,

n

n

,

°

.

.

e

Z

,

n

.

ole N f 0 A is o w t he w e have S i ce A 0 N = 90

the

n

,

o

n

,

°

n

2

2

,

10 1 1]

TH E

-

°

in

re

H E R ICA L

C O O ED I NA TE S

d

°

on

nc

n

e

d by the t d

,

n

.

u a

o

n

e

r

n o les

be t w e e n the i r

are

e,

2

n

ra

o

n

n

o

n

°

,

,

x

s

the i n c l i n a

me a s u

an

e,

a

as

SP

as A ON 90 we have N 0 N 18 0 alre dy e plai e d is t he i li atio f the two grad ua t ed t h i case T h e w e b t ai t he i m por t a t result t ha t tw d t d i ti b tw e e e a t c cles i s a l w ys g g

A , 0 N , = 90

which circles

USE o r

r

a

.

o oub a ques t io m y arise as t o t he arc N N ( Fig Is i t the lesser f t he t w o arcs w hic h w e shoul d aturally take or is it t he arc recko ed t he o t her way rou d t he circle fro m N by d A ? T here are thus tw o arcs toge t her m aki g A of which ei t her m y i o e se se be regar de d as t he i cli a t io W e however re m ove y a m biguity thu ari i g by t he o t io t ha t the i cli atio f two gra d uated grea t circles is ever t o e ceed E 1 If B C CA A B b t h p iti ve d i r cti th r grad uated g r at ci rcle which f rm t h t ri a gle A B C d i f A B C b thei r re pecti ve le h w th at h If B A A B b t i de f t h 1 C C th p iti ve d i e cti ( ) lar t r i a le B t h le th i d e A B re e cti vel y A C f 0 g p p d a g le f AB C re p ctively uppleme tary t ( 2 ) T h ide d i de f AB 0 t h a gle If E 2 8 d 8 b th l f t w graduated ci rcl h w that f t h t w ci r c le if i t h i c l i ati N

n

a

,

,

n

n

an

.

n

o

2

2

,

n

,

n

n

n

a

n

n

n

n

n

n

s n

s

an

ca n

n

c

.

n ve n

n

o

x

x

’

,

.

.

e

e

os

n

e

o

s

e

,

on s on

’

’

an

,

’

ee

e

e

,

s

no

s,

o

s

”

'

n

o

e

x

e

.

.

s

n

e

e no

s an

s

s o

a ,,

, an

s an

n

”

’

s

e

on

n

d

that i f

a,

8

a

sin

r

’

’

on s on

s a re

o se s

s o

2,

o

8

e

2

'

a re

s

e

s

s

,

,

=s

i

e n o es o

o

e

e

s

o

e

.

s

o

n

8, s i n 82 + c o s 8,

n

es of

n

i

:

co s

8,

co s

the

82 s i n s

co s

o

es s

o

s

co o rd i at

the

a re

s o

n

os

e

.

co s e an

e

,

,

’

’

’

’

’

m

82

co s

co s

i t r cti e

n

se

(0 2

a ,

(a , of

on

t h e t wo

ci rcles

)

e

8, s i n 82 s i n

sin

a,

8,

co s

82

co s

si n a z

s rn 6

co s

8 si n

a

=

sin

8,

co s

8,

s

where t h u pper e

an

d

l wer ig refer t o

s

si n

co s a ,

n s

o

m

8,

co s a ,

e

t h e t wo

i ter ecti n

s

ons

.

t e r s e c t i o n s o f t w o gr a d u a t e d gr e a t c i r c l e s L t C d 0 ( Fig 1 5 ) be t w o gr duated grea t circles which i t ersec t i t he t w o diam e t rically o pp si t e p i ts V d V L t N be the ole f C d N t he ole f C A poi t m ovi g alo g 0 i the posi t ive d irec t io crosses a t V i to the p i ti e he m isphere bou d ed by 0 T hus V is describe d as t he a s ce di g o de o f C wi t h respec t t 0 11

O n th

.

e

in

.

”

an

n

e

a

.

'

'

n

o

n

an

o

.

"

n

n

n

os n

B

.

A

.

n

n

n

n

v n

n

an

o n

o

n

’

.

o

.

’

.

e

TH E U S E O F

SP

H E R ICA L

CO O

R D I N AT E S

oi t movi g alo g 0 i the posi t ive di rectio crosses at V i to the n eg ti e he m is phere bou d e d by 0 Thus V is described as t he de cen di g de o f C wi t h respec t t o C C fro m which coordi a t es are m easure d I f 0 be t he origi PN 8 t he d 8 are the coor di ates o f N the d GP ole f C w i t h re pect to 0 A t he a gle betwee t w o g raduate d grea t circles is t he arc betwee t heir oles 1 0) w e se e t hat 90 8 is t he i cli a t io bet wee C n d C we have O V = 0P + P V A p

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'

0 V + 18 0

°

a

thus we ob t ai t he f llowi g ge n eral s t te m e t If 8 be t he coord i a t es o f t h e ole f e gra d ua t e d great circle 0 w i t h res pec t t a o t her 0 the the i cli atio f the t w circles is 90 8 t he sce di g o de f C C has coor di a t es 0 d t he desce di g n ode o f O C h a s coordi ate 90 +

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as is o ft en co n ve ie t w e t ake Q

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o

°

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n

.

,

F ro

.

15

,

0

as t he coordi ates o f the n

.

asce di g ode f C as the i cli atio o f the two C an d l f circl es w e have ( Q as the coordi ates t he o e 90 ) ( f the circle f re fere ce bei g 0 n

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§

1 1]

T H E U SE

or

SP

H E R ICA L

RD I NATE S

CO O

35

ge eral to fi t h posi t i d d irecti f gra d ua t io f o grea t circle with respec t t o a other we m us t k n ow three m f ara e t ers t he seco d circle wi t h regard to the fir t We p m y f i s t a ce be give t he t wo coordi a t es f its ole F t his fi es t he ole d t he n o t o ly is the grea t circle d mi e d f whi h that n ole is t he pole but also t he d irec t io t i which t he graduatio d va ces t he seco d ircle is k w give t he coordi ates o f a pole f t he I f we h d m erely bee great cir le t he d ubt the place f t he great circle woul d be defi ed bu t so lo g as it i u k w whe t her t he give pole is the ole or t he t i l e t he d irecti o f gradua t io will re m ai u spe ified T h e third para m e t er is require d t o fi x the o i gi f t he seco d circle t h gra d ua t i o d i g o de f the seco d circle O we m y be give Q the a s the firs t d also the i cli a t io S tar t i g fro m the origi d thus fi d the asce di g we set ff Q i the p si t ive d irec t io ode T h seco d ci cle is the n e teri g t he positive he m is phere I f we m ake t he tw di e g i g arcs fro m t he ode o f t he firs t con ta i t he a gle t here is o a m bigui ty as to t he e act place f the cir le require d d e f C with r gard t E 1 S h w th at t h a ce d i g C i th de c d i g de f C w ith regard t E 2 S h w b y a fig u r t h d i ffere ce b tw e t w grad a ted great circle which h avi g eq al i cl i ati t t h grea t ci cle f r fere ce h ave t h d i t a ce f thei r a c d i g d e fr m t h re pectivel y 6 d ri gi E f t h a ce di g de f a grad ua ted grea t 3 If 0 b t h l g itu d c i rcle L d i t i cl i ati t a fu dame t al ci rcle d if Q b th c rre p di g qua titie with regard t a ther grea t circ le L det rmi e t h c rd i at f t h a e d i g d e V f L u p L th L t N N ( Fi g 1 6) b t h de t h fu dam e tal ci r cle ON V i t h a c di g b t h d i ta c e N V W h ave L l t de f L u p t fi d i t rm f d Q —Q Fr m f rm la (6) i 1 w b ta i In

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take

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H E RI C AL

SP

TH E U S E o r

CO O

R D I NAT ES

k w we de term i e 8 t h c rd i at tal ci rc le fr m t h equati n

n

n o

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with re pect

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es of

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co s

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Q)

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Q)

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Q —Q '

la t example fi d t h i cl i ati d Q re pecti vely p ified b y Q grea t ci r cle h ave f u d th at t h c rd i at f t h le Q + 2 7O 2 w h ave d h e ce b y 10 E 90

Ex 4 .

t he two

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t h e t wo g

o

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si n e sin e

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(0

t o f c o o r di n a t e s B ei g give t he coordi ates f a poi t w ith regar d to e gradua t e d grea t circle i t is o ft e ecessary to de t er m i e the co r d i ates f t he sam e poi t wi t h regard to a di ff ere t grad uate d grea t circle L t 8 be the rig i al coor d i ates f a poi t P d le t 8 be t he coordi a t es f t he sa m e poi t P i the n e w syste m I like m a er let 8 d 8 be the origi al d tra s for m e d c r di ates o f s m e other poi t P S i ce t h e tra for matio ca t affect t he d is t a ce P P we m ust have t ha t di ta ce the sa m e w hichever be the coord i ates i which it is e pressed d c seque tly 8 ) 12

T r a n s fo r m a i o n

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U SE o r

TH E

SP

H E R ICA L

COO

R D I NA T E S

be t h gradua t ed great circle wi t h ole at N d rigi L t Q Q be 0 to whic h t he oor di ate are t be t ra for m e d the dista e fr m O d 0 res pec t ively f V t he a ce di g ode t he fir t L t be t he i cli a t io f t he f t he seco d circle t wo gradua t ed circles The Q Q are the three para m e t ers which c mpletely defi e i every way t h seco d gradua t ed great cir le wi t h re fere ce t o t he first We h ave w t o selec t t hree poi ts o t t he s m e grea t cir le d such t hat t heir co rdi a t es i bo th syste m s be d irectly perceived T h poi ts we hall h o e are respectively V A It d N is obvious fr m t he figure t ha t as VA VA 90 the coor di ates t he two sys t e m are a s follo w s o f these poi t s i = Q For V = Q ; d ’

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Le t O A

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80 = 9O

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;

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e

.

ubs t itu t i g t he e coord i ates successively i the equatio ( i ) w e have t he ge eral for m ulae f tra s for m a t io Q) ii cos 8 cos ( Q ) cos 8 s ( ( ) cos 8 i ( Q ) si 8 i cos 8 cos i ( i 8 Q) si 8 cos cos 8 si si ( Fr m t hese we d erive cos 8 o ( Q ) cos 8 cos ( a Q ) cos 8 i n ( Q ) i 8 si cos 8 cos i ( Q ) i 8 i 8 cos cos 8 i si ( Q ) f by m ul tiplyi g ( iii ) by cos d a dd i g ( iv ) m ul t i plie d by si we obtai ( v ) d by m ultiplyi g ( iv ) by cos d subtracti g ( iii ) m ulti plied by i we obtai ( vi ) T h firs t set f equatio s d eter m i e the coordi a t es 8 whe 8 are kn o w d t he seco d set d eter m i e 8 are 8 whe k ow A o t her proo f f the fu da m e tal for m ulae f t he tra n s f r ma t io f s pheri al coor di ates m y be obt i ed i t he follow S

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the pole f N N ( Fig Q w he ce A N N P ’

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w e have A VN N = 90 a Q We also see 90

1 7)

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°

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TH E

USE o r S P

H E R ICA L

CO O

RD IN ATES

tha t Q A VN D w he ce 4 N N P 90 Q Th e figure also shows t hat N P 90 8 N P 90 8 d N N I the tria gle NN P w e t hus have e pressio s fo the three i des a n d t wo a gles d he ce fro m t he fu da m e tal f rm ulae 1 f 1 d d w e e uce ii iii iv t he la age t ( ( ) ) ) ( ( ) p Th e essi ty already p i ted t f havi g t hree equa t io s i t he for m ul a e f t ra n s for m ati m y be illustrate d from the grou p ( ii ) ( v ) a d ( vi ) d 8 fro m equatio s ( ) S u ppose t hat w e sought d (v) w e have at o ce i i i Q s 8 cos 8 cos s Q sec Q t 8 e e ( ) { ( )} ) ( the right —ha d side are k w t ( Q) A all t he qua ti t ies is k o w L e t 9 be t he a gle :l 1 8 0 whic h has t his value f i t s ta ge t the ( — Q ) m ust be either 9 or 9 + we —Q d ecide which value is to be t ake fo by equatio ( ii) For as 8 d 8 are al w ays betwee t h li m its — 90 d cos 8 n d cos 8 are both ecessarily positive Th e Sig o f cos ( Q ) m ust there f re be t he sa m e t he sig o f cos ( Q) It is thus ascerta i ed w he t her Q is t o be 9 or 1 8 0 9 f f these a gle will have a cosi e agreei g i n sig o ly o w i t h cos ( Q) T hus t he two e quatio s ( ) Q ) without d ( v ) d eter m i e ( a m biguity d there f re is k ow We the fi d cos 8 fro m ( ii ) A t t his poi t t he i su fficie cy f t w equatio s beco m es a ppare t f though t h m ag i t ude o f 8 is k o w its Si g is i determ i a t e H e ce the ecessi t y f a t hi d equatio like ( vi ) which gives t h e value f i 8 d he ce the sig f T h e proble m o f fi d i g 8 fro m ( v ) d ( vi ) m ight also be solve d thus E quatio ( vi ) d eter m i es i 8 a d thus shows that 8 m ust be or o t her o f t w su pple m e t al a gles It is ho w ever u derstood that 90 :l 8 ii 90 d w e choose f 8 that o e o f t he su pple m e tal a g les which fulfils this co ditio T hus 8 is k o w Q) d E quatio ( ii ) will t he give cos ( a d he ce cos m v w ill give he ce Q is deter i ed wi t hout i s ) ( ( am bigu ity a both its i e d cosi e are k ow a

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i determi at 8 = 90 ati fy t h equati ( ii ) (iii ) ( i v) a d veri fy th a t th e e qu a titi e f t h equati o A a verificati m E 3 ( ii ) (iii ) ( i v ) h w th at t h f t h ri ght h a d mem b er i u ity f th qu are 4 S h w th a t t h E w r itt qu ati ( v) ( vi ) m ight h ave b ce fr m ( iii ) ( i v ) d w at F d i g o d e f 0 A w ith re p ct t O A V i th d T hi i m pl i e that d 8 m y b i t r ch a ged w ith d 8 i f at t h am e ti me Q d Q b each i c rea d b y I f t h pla e f t w o grad u ated gr at ci r c le c i cide t h w E 5 f t h c rd i at cti 8 o d th c 8 g rad u at d great ci r c le th er f t h ame p i t t h ph ere th I t h g e eral f rm ula ( ii ) ( v ) ( v i ) w m ake = 0 i f t h e t w ci r cle = h m d d i t d ra u a t e a e i r cti d 1 80 i f th ey re g rad ua t e d i g I t h fi rs t c a e pp ite d i r cti ’

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’

8

—a

—8

’

,

Q)

(a (a

Q

)

.

co rd i ate 8 h ere ch a g ig b e au e t h rever al f t h e di recti f g radu ati i terch a ge t h p iti ve d gati ve h emi ph re E L t S b a fu dame tal grad u at d great ci rc le d l e t B X be t h 6 c rdi ate f y p i t P with re pect t S L t S b a th er grad uated le with re pect t S d l t 6 x b t h e c rd i a te f it g reat cir c le L t Q de te t h d egree m i ut S a t i t a ce d d ec d mark d i g Sh w de S L t B X b t h c o rd i at f P w ith regard t S th at f t h deter m i ati f B X i term f B Th e

o

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on

x

oo

.

n

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e

.

s

n

e

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e

0

n

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e

Q

0

)

co s

B s i n (A

AO)

co s

6 s i n (X

Q

0

)

si n

5

130

co s

’

or

the d

i

cos

co s

= B si n ’

e term i ati n

on o

B S i n (A AO) 6 co s s in

3

co s

’

8

’

si n

6

Si n

’

B

co s co s

sin

term

s o

(X 30

5

0

’

Q

'

.

AO),

f B, N ’

)

O

cos co s

B s i n 30 s i n (X

Q

0

BO s i n (X

Q

O

’

’

8

co s

s

s

3 s i n 30 c o s ( 7x t o )

3 s i n 30 + c o s 6 c o s 30 c o s (X

f B, k i n

cos

on

o

,

(N

e

s

o

s o

B

co s

e

s

.

,

s n o

o

cos

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s

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e

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es S

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o

1 2 — 1 3] Ex 7 .

Le t

.

y tem ytm w e h ave s

s

s e

a,,

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an

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n

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.

acc r i ext ri r o

e

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s

h ere

p

17 i s

.

ch a

the

n

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si n

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on

s

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e

cos

,

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cos

s

s

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tsid e

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on

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82

(a ,

'

cos

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ge s i n t h e o o d ppo s d t o b e

n

s

on

e

o

n

n

s

)

0 2 ’

es

s

( ii ) ( iii ) ( i v) ,

R D IN A TES

c rd i at f t w tar i t h fi r t rre p di g c rd i ate i t h ec d tar m u t b t h a me i b th y te m

(a ,

cos

o

os

n

s

e

ee fr m t h B t i f w w i h Fi g i ide th e V i t h as s

on s

s

COO

c r i ate t h c ele tial u e vi ewed fr m t h i ter i r ph ere i h w th at t h f rm ulae rem ai u al ter d upp ed t b e d raw a s u u al fr m t h appeara ce

E p d n g as the o and s o 8

.

equati

m th e

o

co

t h e t wo

sin

veri fy thi fr

H E RIC A L

82 b e t h e

8, si n 82 + c o s 8,

si n

SP

’

i ta ce

As the d

.

8,

d

an

s

T H E U SE

n

s

s

n

e

e

o

s

h ere

p

or

the

of

th e

.

o

e

n

.

repre e t a p rti f t h phere ee f m th d de I t ad f Q d 8 w h ul d v i t di g 18 0 + Q d d i m i larl y 18 0 + Q Q T h e e ch a ge 8 d —8 f ma k e alt erati i t h f rmulae (ii ) ( v) (vi ) E f tw p i t If d h w that t h 9 8 8 t h c rd i at e d e f t h grea t ci r c le j i i g th em ri gi b y qua titie d i ta t fr m t h L d L + 180 wh ere u

e

e

ns

s

s

n

°

an

—

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.

no

on

a,

.

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e

an

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17 to

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e s c en

s n

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e

’

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a ,

e

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e

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as

s

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e s

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e

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ro

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e

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n

s S

e o

o

e

n

n

s

°

an

t a t i o n t o L o g a r i th m s If i calculati g the tra s f r m ed c ordi ates t he equa t io s ( ii ) ( v ) ( vi ) 1 2 ) be used as they sta d the t w o t er m s i ( vi ) should be evaluated logari t h m ically d the 8 i take fro m a t able f a t ural S i es Th equa t io ( ii ) de t er mi es o ( Q) a d ( v ) is use d o n ly t o d eter m i e the ig o f Q f this w e ee d calcula t e o ly the logari t h m s f the t wo ter ms the righ t ha d sid e eve whe t hey are f oppo ite sig s fo It is however o fte t hought co ve ie t to e ff ect a t 1 2 ) by t he i tro d uctio n o f m atio f t h e form ulae ( ii ) ( v ) ( vi ) au iliary qua ti t ies w hich wil l m ake t he m m ore i mm e d iately ff d T m t l ga i t h ic calculatio hi be bes e ec t e as m a d a t ed f s y p follo w s d M a gle be t wee 0 d L e t m be a positive qua ti t y 3 60 such t hat i 8=m s M ; cos 8 si ( If M is the s m allest a gle H e ce t M = t 8 si ( a — Q ) A m is which sa t isfies thi M is either M or M + 1 8 0 13

Adap

.

.

’

n

,

n

n

,

o

n

o

a

n

n

,

n

n

e

.

n

n

n

a

n

o

n

s

n

n

,

’

’

s

a

’

or

n

.

ra n s

n

r

n

,

,

x

c

’

on

o

n

n

s

o

n

,

’

n

S

n

n

n

n

n

n

,

an

o

,

n

or

o

n

r

a

.

.

an

an

n

°

n

n

an

°

n

an

n

co

s n

n

co

a

n

,

.

°

s,

,

0

.

s

TH E

SP

USE OF

H ERIC AL

CO O

R D INATE S

[

OH

.

11

f M i t ive we us t c h o se t ha t value which gives M wi t h m p By the sa m e i g T h s l g m a d M becom e k o w i 8 subs t i t utio f t hese au iliary qua tities i ( ii ) ( v ) ( vi ) 1 2 ) these equa tio s be o m e cos 8 ( Q ) cos 8 s ( Q ) cos 8 i ( Q ) m i ( M ) m cos ( M si 8 ) From the la t o f t h e e for mulae 8 is ob t i ed both as t o m g T his value subs t itute d i the t w o i t d ( :l d as to ig o t her form ulae d e t er mi es both cos ( — Q ) d i ( — first gives the m ag i t u d e f Q d t h e ec d gives h T i t s ig or

o

o s

S

as

n

n

s n

o

u

.

n

n

n

x

o

co s

n

,

n

.

,

c

n

’

’

’

’

a

co s

co

’

'

s n

a

s n

a

e

’

n

6

’

s

u

n

e

s

an

>

a n

s

n

n

.

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’

n

a

n

S

a

o

n

e

a

’

’

an

a

s n

'

an

on

s

.

= 75 c rd i a t f a p i t S h w b y th e f rm u l e ( ii ) ( v ) ( v i ) th a t wh e t ra f r med t o a ci rcle f refer ce = = = d efi d b y t h 5 u a titi e c o rdi ate Q 2 1 2 Q 3 I 15 th q

Ex

o

.

1

Th e

.

a

,

e

a

Ex 2 .

s

n

Ex

.

M

an

If

3

.

m = co s P K

an

n

are

n

s

a

o

,

n s o

°

°

e

,

r b lem f E d m h w th at M VP ( Fi g 1 7 ) wh o

o

s

x

.

1 be

d M = N K,

an

d

o

so

en

o

°

’

e

lved with t h h lp e

e

o

2 92 38 a n d L o g m = 9 8 2 7 8 °

o

.

t ri a gle N P K ’

o n

n

o

s

°

If t h e p

.

ua titie

q

= 32 7

o

,

n

e

’

es

n

,

ne

be co m

oo

°

’

f the

au x i l iary

'

.

r d uced meet NN i K h w that b ta i t h f r m u lae ( i ) fr m t h r i ght a gled p

en

n

o

'

s

e

o

o

n

S

e

o

-

n

C H A PT E R I II

FI G U R E

TH E

.

I

O F T H E EA R T H A N D M A P M A K N G

I tr duct ry La titu d R ad iu f cu rvatu re al g t h m er i d i a T h the ry f m p m ak i g th at a m p hall b c f rmal C d iti Th cale i a c f rmal repre e tati M er ca t r pr j ecti Th l x d r m e S t re graphic pr j cti Th tere graphic pr j ecti f th y ci rc le al a ci rcle Ge eral f rm l ae f tere graphic pr j ecti M p i which each ar a b ear a c ta t rati t re p d i g area t h phere o

n

.

o

e

s

e

o

on

a

s

’

o

e

s

o

o

e

on

n

on

o

s n

o

on

o

on

o e

s

e

s

on

o

o

e

n

n

a

o

on

e

e

on

o

o

o

on

o

an

on

h er

e

sp

is

e

so

n

o

a

u

or

s

n

s

o

e

on

n

s

on

on

o

ons

n

o

o

the

co r

s

e

tr o d u c t o r y That t h e earth is globular i for m would be suggested by the a alogous for m s o f the su a d m oo d it is d m st a t e d by g M M fa m iliar fac t s as set forth i books o geog a phy A ccurate m easure m e t s o f the figure f the ear t h are o f fu da m e t al i m por t a e i A t ro o m y d t his cha pt er w ill be d evoted t o t he ele m e t ary parts f this subject as w ell a s to e pl ai i n g how curve d sur faces such as tha t f the ear t h be d e picted fl t sur faces i f m p m aki g t h e art I t is e c essary to e plai that by t he e pressio figure f t he earth w e d t m ea i t irregular sur face diversified by o n ti e t d ocea as w e actually see it but a sur face part f w hich is i dica t e d by the cea at res t d which i other parts m y be defi e d coi ci d e t wi t h t h level t o which w ater would rise at the place i f freely co m mu ica t i g wi t h the sea by m ean s f ca als w hich w e m y i m agi e traversi g the co t i e ts fro m ocea t ocea 14

.

In

.

n

n

n

n

n

n

n

,

e

an

r

n

n

n

n c

n

s

n

a

x

o n o

an

n

.

n

.

“

o

c

,

,

o

a

n

an

,

e

n

n

a

n

a

s

n

ca n

,

x

n

n

o

n

o

n

n

n

as

as

,

n

n

o

,

on

n

an

o

n

n

.

o

n

x

r

o

n

o

n

n

n

n

n

n

n

o

‘

T HE

F IG UR E

THE

or

EA

RT H

AN D

[OH

M A P M A K IN G

.

111

t tu d e I f the ear t h be regar d ed as a s phere the the lati t u de f y s t ati t he ear t h sur face is t he i cli atio to t he pla e o f t he terrestrial equa t or f t he terres t rial radius to tha t statio B t t he true figure f t he earth is n t s pherical It rather appr x im a t es to the s pheroi d f revolutio obtai ed by the rotatio f ellipse abou t i t s m i or a is Th le gths f the s e m i a es o f this elli pse by C olo el C larke i are s give 2 0 9 2 620 2 feet a ( appr i m ately ) 3 9633 m iles 637 8 2 kil m etres 2 0 8 5 4 8 95 feet b ( appr i m ately) 394 9 8 m ile kilo m e t res T h figures i square brackets de ote the logarith ms f t he u m be r s to w hich they are a tt ached If the or m al P N to the earth s su face ( Fig 18 ) m eet the f la e t he equator i N be the se m i a is m ajor the N d O A p 4 P NA d A P OA is ) is t he g e o g (f p hi c l lati t u d e f P i t s g e o ce t i c la t itude 15

La i

.

.

n

,

an

o

’

s

on

on

n

n

n

n

o

n

o

o

o

x

n

n

n

o

an

o

x

-

‘

n

n

a

n

e

.

o

.

n

u

.

ox

o

ox

e

s

n

o

n

n

.

’

n

n

o

n

a

t he equation n

n

,

an

o

.

o

f

.

t he elli pse

coordi ates f a poi t the we easily see t hat th e

x

-

F IG

If

.

an

ra

n r

r

n

o

P

o

f

18

.

be z z g g which 7x is t he e xce t ric a gle be

l

- -

i -

—

1

,

an

d )

t a n x/b,

a

f G e o d e sy C

‘

,

lr

a en

v .

n

n

tan

d

'

’

t a n qb

b t a n X/a

d o n Pr e ss , 188 0 , p 31 9 .

.

,

an

d g

’

n

,

FI G UR E

TH E

T

aki g n

RT H

A ND

MAKI NG

M AP

as u ity we have cos gb cos h cos MN/ m

[O H

.

II I

n

a

’

’

r

v

(

.

’

’

p

r S111 c

7t = ( 1 —

b si n

y

t here fore we

If

O F TH E E A

m

2

e

,

—

)

2

e

Si n

¢

2

.

ake (1

2

6

)

cos Y d log Y are give i the Ephe m eris T h qua tities l g X each d egree f (I A si j> is m ulti plie d by e i X d Y f a s m all error i 4 will m ake appre iable e ffec t X d Y d log Y m y be obtai ed by i s pec t io T hus l g X f the table withou t troubles m e i t er polatio t he accurate T he values f l g i 96 d log cos ¢ bei g add e d to l g X d l g Y respectively w e b t ai l g i qt d l g cos d the ce W e m y ote that l o g X d log Y have a co sta t d di ffere ce As illus t ra t io f the applica t io o f this m ethod we m y t ake t h follo w i g case T h g e gra ph ical la t i t ude o f C a m bri dge bei g 5 2 1 2 5 2 ho w t ha t t he re ductio to be a pplied to obtai the geoce tric d fi d the d is t a ce o f C a m bri dge fr m the la t i t u de is 1 1 earth s e t re whe the earth s equa t orial adiu is t ake as u ity log Y = L gX 9 9 9 7 9 5 99 w e obtai n

X si n

’

3

n

e

o

o

or

n

)

s

.

n

2

<

c

a

n

n

an

.

o

n

.

o

n

a

an

r s n

o

,

an

n

n

o

an

n

n

’

n

on

n

an

,

n

2

n o

)

s n

o

n

n

o

o

co s

an

an

o

r

an

r

o

o

an

n

an

r

n

n

.

n

an

o

n

e

e

a

n

.

n

o

n

S

’

°

,

n

n

'

an

’

n

’

n

n

c

n

o

r

s

n

n

.

o

Lo g

L o g s i n qS

Lo g Lo g

i

r s n (

ta n

Lo g

d)

r co s

¢

r si n

Lo g

If

the 1

d

I

() ’

9 8 9668 0 1

gb

a id

in

could

o

o

J

52

°

< ,b ‘

9 97 8 8

’

52

1

’

30

11

’

22

12

(b

.

.

S

on e n

f

n

n

r c

s

’

,

an

n

t c m put i .

52

°

’

9 9 9 90 77 0

r

r co s

E

r

course also have bee fou d fro m o gb but d w e a d here to t he rule o f usi n g the larger o f qua ti t ies see p 7

r

t wo

cos qb

’

9 8 95 7 5 7 1

)

r

r sin

4

0 1 0 7 5 7 90

’

’

Lo g

97 8 7 2 5 34

)

’

Lo g si n

Lo g

co s

g

i

g th e

ve s

ta b

a

r du ti e

.

.

,

c

l

on

i n M o n t h ly N o ti c e s , R A S

e

.

o

f th e

l titud a

e an

d lo g

r

.

.

.

vo l

.

xu n

.

p

.

1 0 2 fo r

15 16]

THE

-

Ex 1 .

Us

.

i

F I G URE

g C

n

larke elem t ’

s

en

d)

ta n t he the

fig ur

TH E E A

or

RT H

fi gu re

fo r t h e

s

AN D of

M A K IN G

M AP

th e

47

earth h w that s

,

o

[9 9 970 35 2 ] t a n

’

cl d i b racket r pre e ti g a L garith m d h w th at ge graphical latitu de f G re e w ich i 5 1 2 8 38 i t ge ce tric latitu d e i o se

es en

s

n

o

o

°

e

5 1 17 11 '

Ex 2 .

s n

n

n

If po

wer

f

s o

hi gh er tha

e

n

the

s

X = 1 — § e2 — i Y= 1 + i

h w that t h e tab l c o cer ed ma y b e c o mputed fr .

n

’

”

d

a re n

°

s

,

s

an

o

S

o

as

n

s

”

.

Ex 3

o

S

.

e

2

2

e

fo r L o g X om

es

o

n

Lo g X

9 99 7 7 8

Y

0 0 00 7 4

lo g

ec

on

co s

2 d>,

co s

2 d)

an

d Lo g

co s

000 7 4

co s

‘

,

s

o

.

000 7 4

‘

eglected h w th at

Y

2 d)

fa r

so

as

lace

fiv e p

s a re

.

a l o n g th e m e r i di a n t a t y poi t i T h curvature o f t he earth alo g a m eridia the curva t ure o f the circle which oscula t es the ellipse at that poi t the ellipse I f a cos 9 b si 9 be t he coor di ates f a poi t t he the equatio f t he orm al a t t hat poi t *

16

R a di

.

u s

o

f

c u rv a u r e

.

n

e

n

an

s

n

n

n

n

,

o

n

n

on

n

n

o

.

n

is

a w s in 9

the la t itude the m ajor a is

and

fo r

d )

by

—

co s

9=

(a

2

— b2

) si n

9 co s 9

or the a gle which t he or m al m akes w i t h n

,

n

x

ta n

(

5

a

t a n 9/b

.

ce t re o f curvature is t he i t ersectio o f two con secu t ive or m als D iff ere t ia t i g ( i ) w ith regard to 9 w e see t hat the o o d i n a t es o f the ce tre f curvature m us t atis fy t he equa t io cos 9 by si 9 ( a b ) cos 2 9 t he c d ( ii ) w e have f S olvi g f a n d y fro m ( i ) ordi ates o f the ce t re o f curva t ure i 9 b b a cos b 9 a ) ( / ( ) / y t he r d ius f curva t ure we the fi d an d f Th e

n

n

n

n

.

r

n

n

c

o

or

or

an

a;

o

n

a:

or

a

a

2

ter m s

o

,

(

sin

”

9

b

2

co s

,

2

the lati t u d e p

H e n ce

a

2

s n

3

n

n

o

f

?

2

3

?

p

in

?

2

n

n

or

n

s

a re

n

n

a

w e see that

2

b

2

if

(b

cos oS) be the d ista ce betw ee t w o poi t s

?

s

si n

?

1

()

a

2

2

n

n

n

on

F IG UR E

TH E

t he sa m e

m

radia s are

eri dia

b g

n

RT H

EA

MA K ING

M AP

AN D

[

CH

111

.

whose geographical la t i t u des e presse d respec t ively w e have

in

x

n

d

an

THE

or

,

2

a

b

2

? cos t q) 1 It easily foll ws that i f powers o f the eccen trici ty above the seco d are eglec t e d we have as n a ppro i m ate value o f the arc bet w ee t he la t itu d es (j a d gb

(b

si n

2

2

2

n

.

’

’

an ’

n

n

n

an

o

n

z

n

n

z

z

n

n

n

n

on

z

n

n

,

on

an

n

’

on

.

u

n

’

,

an

ns

n

,

.

e

z

.

0

n

n

,

n

n

n

o

,

n

°

.

n

n

o

z

n

o

,

z

n

n

n

or

’

n

n

z

n

an n

o

no

r

.

.

.

n

n

n

n

n

n

n

r n

n

.

e

n

z

n

,

n

n

§

R IG H T

34 ]

A S CEN SI O N A N D

orm ulae ( ii ) ( iii ) ( iv ) ( v ) ( vi ) equatio s

f

,

,

of

,

,

D ECL I N A TI O N

we have the

12 ,

d

esire d

n

c o s a si n z

=

co s

cos z

=

s i n si n

)

8 — Si n

si n

¢

a

8+

f

S(S

—

d

oos

8

cos ¢

co s

8 c o s (S — a )

co

a

)

the equivale t grou p

d

an

— S (

= c o s 8 si n

— s i n a si n z

n

— si n S (

— a

(S

a

co s

)

co s

8 = si n

a si n z

= cos 8 cos 4) c o )

— si n

s z

t

c

c o s a si n z

the equa t io s ( i ) we n calculate the e i t h dista ce n d a i m uth w he the d ecli atio a d t he hour a gle (S — ) are k ow d co versely by ( ii ) w e c n fi d t h e d ecli atio d t he hour a gle whe n the e i t h dis t a ce d the a i muth a e k o wn For a deter m i a t io o f the e i t h dis t a ce w he t he hour a gle d t he d eclin atio are k o w the followi g proce ss is very co ve ien t Th a gle sub t e de d at t he star by t he a e j oi i g e ith d t he pole is called t he p a l l cti c a g l e T hi w e the shall d e o t e by ) d f its deter mi n a t io we have fro m the fu da m e t al for m ulae ( l ) ( 3) 1 t he followi g equation s i w hich h is writte i s t ea d o f ( S ) f t he hour a gle By

ca

n

n

n

z

n

n

an

,

n

n

n

a

z

n

n

n

n

n

z

a

n

n

n

a

n

an

n

an

z

r

.

n

n

n

an

n

n

n

n

z

e

.

n

z

n

n

n

n

n

n

r

an

ar

n

an

7

n

or

,

si

n

co s

W he n h dis t an ce

a

n

n n s

.

n

n

n

,

,

n

n

n

or

a

n

si n z

=

n

si n z

= si n

si n

p co s ¢

co s (

n

n

h

8—

co s

¢

si n

8 co s h

are k ow the par llactic a gle 7 d t he e i t h A si d n both be fou d fro m these equatio n s cos ct are bo t h lways posi t ive i t f ll w fro m t he seco d equa t io tha t ; d i have t he sam e Sig T hey are bo t h positive t o the w es t f t he m eri d ia d ega t ive t o the eas t It is o fte d esirable t o m ake t hese calculatio s by t he hel p We i t rod uce two e w a gles m n d f subsidiary qua t i t ies by the co di t io cos cos 4 i h i cos m i t i cos 4) cos h Si m d m which satis fy t hese If m be a pair o f values o f equatio s they w ill be equally sa t isfied by 360 n d 1 8 0 m an

2

d 8

n

n

a

,

an

o

n

t

n

an

o

n

n

.

) s n

S n r

n

n

an

°

n

a

n s

s n n

0

n

n

n

.

s n n

o,

n

.

n

n

an

n

n

n

n z

n

n

s

n

o

S

.

,

o

z

an

1

n

ca

a

7

n

o

an

°

o.

R IG HT

92

I t is a w e use airs p

m

at t er f i differe ce whether i the subseque t w ork — 18 0 m T aki g o f these t w o 0 3 6 or m m we have by ubsti t utio i ( iii )

o,

n

as

n

o

o,

cos

i

n

one

n

8 ( + m)

a n n Si n

1; s n 2

n

.

s

,

cosz

si n

°

o

,

n

n

n

n

o

°

n

D E C L I N AT I O N

AN D

ASC E N S I ON

n

i cos ; si cos ( 8 m ) T he e equatio m y also be writ t e thus n z

7

s n n

n s

s

a

n

tan n =

ta n

z

c o t n se c

(8 + m

t

(8 + m

se c

z

n

co

( vi )

.

Fro m the first o f these ) is fou d d the the seco d gives Of cour e coul d also be fou d from the firs t f ( v ) but i t is al w ays f fi re erable t d a gle fro m p its t a gen t rather tha i t s cosi e n

1

an

n

s

n

2

z

.

n

o

,

o

n

an

n

n

n

n

(s

or mulae ( iv ) n d ( v ) m y be ob t ai ed at o ce geo m etrically For i f Z L be pe rpe dicular to N P i Fig 2 9 w e have N L = m d Th e f

a

n

a

n

.

n

n

an

.

ZL

90

°

n

Fm

.

29

.

.

It is plai fro m equatio s ( iv ) t hat as a d m depen d o ly o t he la t i t ude d the hour a gle they are t he sa m e fo sta rs o f all decli atio s It is there fore con ve ie t to calcula t e on ce fo all fo a give observa t ory or ra t her fo a give latitude a table by which f each particular hour a gle a t y s t atio t hat la t itu d e t he values o f m d L o g cot u c a be i mm e di t e l y obtai e d E 1 V er i fy th at t h equa ti n

n

an n

n

n

.

n

n

n

r

,

x

.

e

.

.

t a n ”=

u derg n

°

o

n

n o

n

n

an

n

,

an

n

on

a

n

r

or

3 60

n

n

r

r

n

n

on s

c o t n se c

cha ge whe n

n

m

(8 + m )

an

d

n

an

z = s e c 17 c o t

d ta n

cha ged re

are

n

s pe

cti vely i to n

l 8o + m °

.

an

d

.

et rm i e t h e ith di ta ce d parall actic a gl f t h tar b e i g + 38 6 1 Cyg i wh e it i 3 E i t d ec l i ati f t h mer i d i a d t h latitud f th b erver b e i g 5 3 Fr m equati ( i v) w fi d m = 2 7 4 3 d L g t = 9 6676 ( ) H e ce = 34 8 + m = 65 5 2 d ( v i ) n = —4 8 Ex 2 .

D

.

e

s

e

o

o

ons

°

’

e

n

n

n

an

e

an

z

11“

o

e

n

n

°

n

°

z

,

n

s

on

e

o

e

s

°

n

°

n

s

n

an

e

o

.

e

s

n

'

°

an

o

co

n

°

n

.

n

R IG H T

AN D

AS C E N S I O N

D E C L INA T I O N

ecli a t io H ere the four qua tities co cer e d are h 8 d t he for m ula is t here fore cos = i 4 i 8 cos 4 cos 8 cos h d 8 co s t a t d su pposi g h D i ff ere tiati g i 8 i A cos s s cos 8 cos ) h) A t si ( 4 t f the e fficie t f A d) w e obtai d subs t i t uti g si n cos A4 sec a Az O f course t his m ight have bee obtai ed direc t ly from for m ula = = m A 8 O A 0 as j ust give by aki g 2 ( ) A a other illustra t io a n d o e i volvi g t he parallactic a gle we shall deter m i e w he the parallactic a gle o f a g ive s t ar beco m es a m a i mu m i the course f t he d iur al rota t io Th e co ditio s are tha t while I; d 8 are both co s t a t h d a shall vary i such a way t h t t h e shall be cha ge i 7 T h e for m ula i volvi g 8 ) h is i e A 7 m ust va i h d

n

n

n

.

n

n

,

,

an

n

n

n

s

n

8=

cos

co

n

t n

si n

n

,

,

n o

n

n

,

n

er

.

t a n qt

n

n

a

n

n

an

(

7

.

o

n

n

.

n

n

n

.

n

n

n

x

n

h

,

n

n

o

n

n

n

s

,

.

n

,

,

c

co

)

n

n

n c

or

a

z

n

.

an

z

n z

an

)

n

an

n

n

) s n

S n

z

,

h + si n 8 c o s h

.

z

an

n

r

n

7

,

,

.

i ere tiati g w e have i h A co t cos i h s 8 7 ) h 0 ( n d as the coe fficie t o f Ah m us t va ish cot i 8t h fro m whi h we fi d cos = 0 d the star m ust be o t he pri m e vertical I t his w e have a t her illustra t io o f t hose e ce ption al c a ses i w hich though t hree f the varia t i s are ero the for m u l ae do t require that the other three varia t io s shall al o be ero D ff

n

n

n

7

a

s n

n

n

n

c

,

a

,

s n

,

an

an

,

n

.

n

n o

n

on

o

n

x

z

n

no

s

z

s

i ere t ial for m ulae are specially i s t ruc t ive i poi ti g out how observatio s sh ould be arra ged so t ha t t hough a s m all error is m ade i t he course o f the observatio the e iste ce o f t his error shall be as li tt le i jurious as possible to the resul t that is sought S u ppose f i sta ce t he m ari er is seeki g t he hour a gle f the s i ord er t correct his chro o m eter W ha t he m easures is t he alti t u d e f the s n B t from re fractio n d o t her causes whi h en t irely obvia t e t here will be a sm all error S kill i t he al t i t u d e d co seque t ly i the ze ith dis t a ce T h e observer m ea s ures the e i t h dis t a ce as d co n cludes that the T h e d ff

n

n

n

n

n

n

n

n

n

x

,

n

n

.

,

u n

or

n

n

n

n

o

o

c

n

n o

n

n

,

u

.

n

.

u

n

a

ca n an

n

n

z

n

n

n

n

z, a n

n

.

o

§

R IG HT

35 ]

A SC E N SI O N

A ND

95

D E C L INATI O N

hour a gle is h B t the true e ith dis t a ce is + A i e A2 is the quan tity which m ust be adde d to the observed e ith dista ce to give the true e n ith dis t a ce Th e t rue hour a gle is there f re o t h bu t so m e slightly d iff ere t qua ti ty h A h where A h is the correc t io t b be a pplie d t o h so t ha t A h is t he qua ti t y w sought T h e for m ul a co tai i g o ly t h e parts d 8 h is i cos si S 8 cos 1 cos 8 cos h 4 d regardi g d d 8 as co sta t D i ffere t iati g this A cos d cos 8 si hAh si i h cos 8 i si d substi t uti g i cos ¢ Ah A w he ce sec 4 cosec A Ah Th e follo w i g is a geo m etrical proo f o f this for m ula m oves fro m P t o P ( Fig 30 ) abou t the pole N If the s P P bei g a very s m all arc its e i t h d is t a ce cha ges fro m Z P '

n

u

.

n

z

n

z

z,

.

z

n

z

n

n

n

no

n

n

.

n

o

.

n

,

,

n

,

.

n

n

n

n

z

n

2,

n

)

n z

n

z

) an

n z

,

,

s n

,

,

a

)

n

n

n

n

s n a

2

,

.

)

s n a

n

),

()

an

n

an

n

z

.

n

'

u n

’

to

,

.

n

,

n

z

n

n

A A NP P be perpe dicular to ZP A = = P P i NP A h s n d P are both TP 4 n d A ZP T 7 A cosec 7 w he ce i f N K is pe pe d icular to Z P w e shall have t hat t he ra t e f cha ge f t he A = A h i N K by which we lea e ith dista ce o f the s u w i t h respec t t o the t i me is proportio al ZP

'

.

’

’

If P T

n

,

’

’

1

a

2

7

r

n

,

’

z

z

n

s n

rn

,

n

n

s

z

,

an

n

o

n

o

n

RIG HT

96

A SCE N S I O N

AN D

D E CL INAT I O N

to t he Si e o f the pe pe dicular from t he pole through the We have also n

r

n

su n

.

N K = si n Z N si n

si n

the vertical circle

on

cos d si

(a

)

Ah w hen ce as be fore sec q!) co s ec a A z Th e o bserva t io n shoul d be so ti m ed t ha t

n a,

.

shall be as s mall as p ssible f the the error A will have the s m allest d m i f t h ossible e ff ec o the eter ati e hour a gle I t ollows t f p H e ce the prac t i al rule that shoul d be ear 90 or so well k o w t o the m ari er t ha t f t he de t er m i atio f the t i m e the al t i t u d e o f t he s shoul d be obser ve d w he t h s is o or ear t he pri m e ver t ical d oes t co m e t o the pri m e ert ical the s malles t If the s value f A h/A is e 8 By lvi g t h f rm ulae 1 d Ad h w h w E (3) f A 8 A b d ed uc d t h f r m ulae ( 6) f th b E 2 S h w g m t ricall y th at i f t h a u med d c l i ati err e u t t h ex t t A 8 t h err r the ce pr d uced a determi ati f b rvati f th u e ith di ta ce wi ll b t h h u r a gle fr m o

n

or

,

z

n

n

o

n

.

n

c

or

n

n

n

on

°

n

a

c o se c a

n

n

u n

no

so

.

.

n

o

or

can

x

on

eo

o

.

.

o

s

n

o

‘

o

e

e

e

se

o

an

on

)

o

o

S

,

n

on

o

e

on

’

e

on

n

s

n s z n

su n

o

e

n

8 A8

.

er wh at ci rcum ta c i t h cha ge f e ith d i ta ce f a i r al m ti pr p rti al th r ugh ut t h d y t i t ch a g f

Un d b y t he d u n .

e

o

e s

o

c o t 17 s e c

Ex 3

ss

n

o

e

,

2 an

,

.

e

en

e

o

,

.

e

o

e

v

s e

z

o

x

u n

e

.

u n

e

n

n

n

o

.

s

es

n

n

e

s

o

z n

s

n

o

tar o e h u r a gle ? t t whe ce i d thi s m u s t b c W h ave fr m ( 2 ) A /A h = d) t h eq a t r d th ta t d t h e b erver m u t b tar mu t b c m u t b e a equ a t ri al tar b erv d e ith Ex 4 I f t h e h u r a gle i b e i g d eter m i ed fr m f a c ele ti al bj ect f k w dec l i ati S h w ge me t r icall y th at a d i ta c err r t mall err r A d) i t h a u m ed latitud e gt wi ll pr d uc e ¢A¢ i t h a i m uth i t h e h u r a gle wh ere th at thi err r wi ll ge rally b f l ittle c eque pr vided Sh w al t h o bj ect b e ear t h pr i m vertical T h t ri a gle P S Z i f r m d fr o m t h p lar d i s ta c e P S th Fi g 3 1 e ith d i ta ce Z S ( = ) d t h c latitude P Z Th h d m i t t h f t e aralla ctic a le e a ti ve c a u it ea t r i ia b o i p g ; g T h e t ri a gle P S Z i f rm ed fr m t h e p lar d i ta c e P S e ith d i ta ce Z S d t h c latitu de P Z d> A t ) D raw Z M SZ d S Z very d S L perp d icu lar t SZ th e cl o e to geth r b t S Z = SZ w m u t h ave SL = Z M L SZZ i t h a i m uth At th at SL = Z M = L PSZ = c ec q d SS = SL c + o ) s

on

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R I G HT i re ti ati g co c b t ai

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A S C E N S IO N A N D D E GL I N ATI

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here f e i f x b t h ch a ge f max i m u m a i m uth T

o

ta n A

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of

si n

a i m uth z

2

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.

in t

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o

et

m t h e mo m

n

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sin

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[ Math

.

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t i m e o f c u l mi n a t i o n o f a c e l e s t i a l b o d y A t t he m o m e n t o f u pper cul m i atio 2 9) the right asce sio n f the b dy is the sidereal t i m e T h e proble m o f fi di g t he ti m e o f u pper cul m i atio re duces there fore to t he disc very f t he right asce sio o f the body at the m o m e t whe it crosses t he m eridian T H E T I M E o r A STAR s U P P E R C U L M I NA T IO N I the case o f a s t a the com putatio is a very simple as the a ppare t right sce sio al t e s very slo w ly we e ; f al w ays fi d it by i s pectio fro m the tables d so have ca sidereal t i m e f u pper cul m i atio n a t o ce t h For i sta ce su ppose we seek the t i m e f cul m i a t ion o f A ctur s a t G ree w ich o 1 906 Feb 1 2 w hich fo t his particular d f m ur ose is co ve ie tly recko e ro a are t oo o Feb 1 2 pp p p t o a ppare t oon Feb 1 3 W e fi d i the e phe m eris f 1 906 h m t hat t he R A a t u pper cul m i atio Feb 10 is 14 11 22 4 2 i 1 0 days ; d there fore a t cul mi atio o I t i creases m h Feb 1 2 t he R A is 14 1 1 O tha t day t he si d ereal ti m e m at m ea oo f G ree wich is 2 1 2 6 W e t hus see t ha t A rcturus will reach the m eridia at m m m l 6 44 2 4 + ( 14 1 1 26 f si d ereal t im e a ft er m ea oo o Feb 1 2 We t ra s for m t his i t o m ea t i m e by the t ables give i th au t ical al m a ac m h l 6 15 5 7 ‘

36

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cul m in a t io o f A rc t u r us there fore t akes place a t m Feb 1 2 16 42 I t h case f a m ovi g bo d y such as a pla et or the m oo whose right asce sio c h a nges ra pi dly fro m ho u r to hour w e f roceed as ollows p L t t he right asce sio f t he body be at t hree c o t t f w hich the tables g ive the calculated e u t i e e pochs t values d such t ha t cul m i atio occurs be tw ee t d t T he t aki g either f t he equal i terval t t or t t as the u i t f t ime d su ppo i g cul m i a t io occurs t u its a ft er t we h a ve by i terpolatio fo the R A at cul m i atio Th e

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For e xam ple l e t it be propose d t o fi d the ti m e w he t h e cul m i ates a t t he L ick O bservatory M ou t H am ilto m oo m T he lo gi t u d e is here 8 6 D e 2 5 1 90 6 C ali f r ia t i m e f cul m i a t io the G ree wich a d i f 0 is the local m e 0 m ea ti m e is 8 6 2 5 t he sidereal ti m e t Th e e phe m eris sho w s t hat o n D e m d the m oo s R A oon is 1 8 l 2 G ree w ich m ea m at 0 to 3 3 at 23 d i t is v ries fro m 2 1 9 also see that the cul m i atio at G reen w ich takes place a bo t m G M T I the followi g 8 the m oo s R A i creases 8 22 m about 1 5 ; he ce cul m i a t io will take place at L ick at abou t m m 8 37 loca l m ean ti m e or a bout 1 6 43 G M T Th e por t io o f t he tables to be e mployed i n the accura t e ca lculatio is there fore as follo w s n

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36—3 7]

A SC EN S I O N

A ND

D E C L I N A TIO N

7 1 7 103 hours

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the cul m i atio f the m oo at L ick t ook place at m m 1 6 4 3 1 5 7 G ree w ich m ean t i m e or a t 8 36 local m ea ti m e n

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tt i n g o f a c e l e s t i a l b o d y T h e ti m e o f isi g or setti g f a celestial bo dy is m uch a ff ec t ed by re frac tio P o tpo i g t he co si d er tio f the eff ect to a l t e c h a pt e ( V L ) we here give the for mulae f o f re fra c t io fi di g w hen a celestial body at m ospheric i fl ue ces a pa rt is the hori o i e 90 fro m the e ith 37

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D E C L IN A T I O N

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t t tu d e a n d l o n gi tu d e For certai cl a sses o f i vestigatio w e have to e m ploy yet an other system o f c o ordi ates o t h e celes t ial sphere J ust a s the e quator has fur ishe d the m e a s f d efi i g the right asce sio a d the decli a t ion f a star so the ecliptic is mad e the bas is f a syste m o f coordin ates k ow as celestial lo gi t ude d latitude We e mploy i t his e w syst e m the am e origi as be fore T h e first oi t f A ries is t he origi ro w hich lo gitude is to be T f m p m e sured f t h e m easure m e t is t be that o f d t he directio the appare t a ual m ove m e t o f t he s alon g the ecliptic as i dicated by t he arro w head o Fig 3 5 A great circle is dra w fro m t he le K o f the ecli ptic t hrough the sta S d t he i tercept TS this great circle bet w ee the s t ar d t h e eclipt ic is that coord i a t e w hich is called the l ti t de o f t he t ar Th e lati t ude is posi t ive or egative accordi g as the s t ar lies i the he mis phere which co t ai s t he ole or t he a t i o l e o f t h e ec l i tic m Th the ecli t ic f ro the origi n T to T e e p p the foot f t h e per pe dicular is called the l o gi t de w hich is 38

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s

o

n

n

n

o

n

n

a

n

,

n

n

n

.

n

n

on

n

,

,

n

u

,

§

RIG HT

38 ]

A S C EN SIO N A N D

D E C L IN A TI O N

the seco d coordi a t e T his is m easure d rou d the circle fro m 0 to so that i f t h e righ t c e si obj ect f the ecliptic is i cre sed its lo gitu d e is also i crease d Th e reader will f course observe t hat the m ea i gs o f t h e words latitude d l n gitude as here e xplai ed i their a s t mic l sig ifica ce are quite di ff ere t fro m the m ea i gs o f t h e sam e w ords i their m ore fam iliar use with regard to terre trial m atters I t is usual t e m ploy the letter A t o e press t o o m ic al lo gitu d e n d B t o e press ast ro o m ical latitude thu s n

'

n

n

.

°

as

n

a

n

n

on

.

o

an

n

on

.

o

an

o

n

n

n

a

n

n

n

ro n o

n

n n

n

s

o

.

n

T T = 7t

d TS =

a re

equa t or ecliptic

x

a

an

Th e

x

an

of

d

as r

n

n

,

S

.

the solstitial colure L H i t ercepted bet w ee t h e the ecli ptic i s equal to the obliquity f t he n

n

o

.

F IG

.

35

.

be the R A d decl f S t he t he for m ulae f t ra s for matio are obtai ed either fro m the ge eral for mulae f a d fo the 1 2 or directly fro m the tria gle S E N ( Fig d lo gitu d e w e have the equatio s d e t er m i atio o f the latitu d e i 8 si cos 8 i si 3 cos If

a,

8

.

.

o

n

,

n

n

n

B

si n

X = si a

n

r

n

to s n

n

o

.

an

n

cos

or

n

n

n

an

n

n a)

in 8 +

S n a

co s w co s

8 si n

a

cos 8 cos 7x cos 8 cos d 8 are k n o w by w hich w e d e t er m i e 8 a d X whe It is ge erally easy t o see fro m the ature f t he proble m w hether t he lo gi t ude is greater or less t ha Whe t his is k o w o n e o f the l a s t t w o equatio s m y be d ispe se d w ith W e c a m ake these equatio s m ore co ve ie t fo log ith m ic a

,

ca n

n

n

n

n

n

a

an

n

n

n

n

a

n

.

o

n

n

n

n

n

.

n

n

r

ar

n

R I G HT

A SC ENS I O N A ND

w ork by t he i n tro duction

[

D EC L I NAT I O N

CH v

au iliary qua ti t y M A S T L so d we have that t M c sec t 8 i B w ) cosec M si 8 i (M B i A i 8 cos ( M w ) c sec M cos B A cos 8 cos Th e for m f t he equa t io s shows t hat a cha ge f 1 8 0 i n t he t a ff ect t he resul t d pte d value f M d oes E t h a gle subte d ed at S by K N I f we re pre e t by 90 w e have fro m D ela m bre s f r m ulae cos ( 4 5 J; ) cos } ( E A) s ( 4 5 5, B ) cos {4 5 l ( 8 4 i J E 4 s 5 4 cos si A 5 8 5 + B ( ( ) ) } ) } ( ; ) ( { l J m) l cos ( 4 5 si ; (8 } ) A( E 7x) i ( 4 5 } B ) i {4 5 cos 3 (E A) i ( 4 5 4B) i 45 —} ( 8 w ) } i ( 4 5 } ) by which A d B well as E be d e t er m i ed I f it be require d t o solve t he co verse proble m a m ely to d eter m i e t he R A d lo gi t u d e are d decl whe the lati t ude give we have by tra form atio f ( 1 ) i 8 = cos s i n B + si w c B i X cos 8 i si si B o cos B i 7x cos 8 cos cos B cos 7x E d dec l i ati le f t h 1 S h w th at t h r i g ht a ce i f th ecl i ptic re p ctively right a ce i 90 d th at t h d d ecl i ati f th l ti E 2 If R th d decl f t h p i t f t h e l i ptic wh e 8 l gitu de i A h w that o

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r ve th at = i ( ) ta (t 8 E 4 T h ri g ht a ce i f O ri i i d th b l i quity f t h ec l i ptic i 2 3 +7 d latitu de f t h tar re pecti vel y 8 7

t he

ame l

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t a n 82

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itude

R IG HT A SC EN SI O N A N D D ECL IN AT I O N [C H V h t aralla ctic a l a i t h r i h th a t a i ve la titu e f S w f d p g g p d b cal culated f t he y h ur a gl b y t h f rm u lae qu a titi .

ta n 8

ta n

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co t

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)

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o

.

or an

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on

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= + s i n qt t a n h ,

o

= — si n h

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+ si n

cos

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cos a

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( ) se c

n

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= + se c ( ) s i n 1

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cos

re p cti vel y t h d ecl i ati d t h h r a gl f a tar d e ith d i t a c b ta i t h f l l wi g f rm u lae b y which i t a i m uth th at latitude t h val ue f b ea i l y det rm i ed wh e f ) h m d t e fi e i t la t exa le c rre i h k w d d ) p p g ( Ex 7 .

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latitude 5 2 determ i ed b y b rvati ti me f b ervati h h ur rm ul a i appr x i mat l y an

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.

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of

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b latitude qt f t h p le tar which at t h f t h al titu de d th at t h f d a p lar d i ta ce p gle h E x 10

h w that

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f t h e fo m °

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ht ¢ pc h+5 i l p i e ith d i ta ce whe a ta 11 S h w th at t h h u r a gle h d th E d e W e t m y b f u d f r m t h equ ati i s d e E as t = i h c 8= 1 i h c i 8 = si ¢ o 8 s ; ; ; qt d t h l wer S i gn i t h e latte r b y u s i g t h e u ppe r ig i t h f rmer ca Fi d t h e fi r t d ec d d i ffere ti al c efficie t f t he e ith E 12 d i ta ce o f a tar w ith res pect t o t h h u r a gle W ca i ve tiga t thi s ither fr m t h fu dame tal f rm u lae ge o m et ricall y f ll w Fi g 3 6 rth po le Z t h e ith P t h e tar I t h ti me d h t h tar N i th h m ved t H w h ere P E i perpe dicu l a r t NP d NH If P Q b e pe rpe d icular t Z H th e =a —

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dz = H Q = H P s i n ”= c o s 8 s i n q dh =

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.

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s n

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(8

n

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(8

r ula f A r ctu ru a t t h

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e an

n

s

z n

,

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ua titie u ed parallactic a g l ) c c (8 (

co t z co t

o

are

s

e o

n

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s

si n

on

s

an

o

’

co s

:

q

’

c o s 7)

z

’

o

s i n 7)

.

z = si n

h w th at w ith t h d e term i i g ) t h e s tar S

.

n n

Ex 9

(8

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an

a

n

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z

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c ec os

z dh,

s

§

RIG HT

38 ]

ec with re pect

T o fin d t h e

fo r

u

n

d

ad ia

s

AND

to

en

i

an

ss

111

D E CL I N AT I O N

i er ti al co e ff cie t w h d a u mi g th at

d d ff

on

s

A SC EN S I O N

n

n

i ere tiate

d ff

e

an

a

d h

bo

a re

ab ve th expre ed i d z/d h

n

as

o

ss

n

n s,

— co s

¢ cos a

— co s

¢ cos a

FIG

dh

’

8 c o s n c o se c

cos

.

36

z

.

.

a tar ex c ed t h latitud h w that t h m t rapi d ra te f ch a ge i e ith d i ta c b y d iur al m ti i equal t th i f t h d ec l i ati tha t h latitude I f t h d c l i a ti b le h w that t h m t rapi d rate f cha ge i e ith d i ta ce i equal t t h i f t h latitu d e E 14 If ith d i ta ce f b j ect at t h h r a gle h b th d if b th e ith di ta c f t h ame bj ect at t h h ur a gle h which i very ear t h h w fr m E 12 that E x 13 .

If t h e dec l

.

os

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co s n e

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cos ne

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—z

1 5 (h —h0 )

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aki g A — d i h av fr m t h la t qu ti M

s

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c

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B

D E C L I N ATI O N

AND

AS C E N S I O N

(hi

ho) + 3 (i

n

( k2

720) + 3

h 0)2 7

A (fin

ZO +

n

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(h

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r

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2

0

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which pr ve t h th e rem T hi f rm la i u efu l wh e it i d e i ed t b ta i t h b t re ul t f r m a ri f e ith d i ta ce t ke i rapid ucce i 16 S h w th at i f t h h u r a gle f a tar f d ec l i ti E 8 b h d h wh whe it h t h a i muth it h t h a i m uth 180 + t h b f u d fr m t h eq u ati latitude I o

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rth latitude 4 5 t h greate t a i muth attai ed b y f th P r ve that t h cir cump lar tar i 4 5 fr m t h rth p i t f t h h r i tar p lar d i ta ce i [ Math T r i p ] fi d th latitude i f t h l c al idereal time b E 18 Sh w h w t tar h ave t h am e a i muth b rv d a t which t w k w tar h ave t h a im uth are T h h u r a gle h h a t which t h t w k w d ( p 3) E x 17 .

In

.

o

’

s

s

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se

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s

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n

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n

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n

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i

n

n

s

s

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h

c o t a si n e

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e

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n

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s

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on e o

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o

.

e

.

o

o

n

z

o n

e n o

o

o

e

e

s

s

o

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e

h = ’

s

z

s

o

j

8 + si n

e

1 ta n

8 + sin

p

h,

t

h

c co s

’

(

e

.

s

t a n

co s ( )

s

o

co s

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a

’

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g

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’

— ta n

(h

'

’

8

s in

h

h)

im l ta e u sl y b rved a t t w a e eri ia at a t im e whe o th g pla ce t h decl i ati f th P r ve th at i f j) b t h latitu de f i 8 f th e f th i r latitude i appr x i ma te l y place t h di ffer c Ex 19 .

o

Tw o

.

se

o

e

on

n

s,

al titude i g hb u r i

n e

e su n

o

e

en

of

s

e o

the

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e

B

an

s

e

8

e

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l

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itude

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an

h w that

d

o

a)

t he

ob

if

a

s

u

f th e

e

is the

l i quity si n

o “

1

(s in

Si n

e

al titude i t h pri m cl i ptic t h latitude f t h ’

e

n

e

,

si n

on e o

o

Bsi n

n s

a)

o

n

xam ] vertical L i t plac e i

[C o l ] Ex 2 0

n

o

s

Bco s

d B+ AB, a re m m d n

(

,

s

ABc o s

,

s on

n

s

su n

L /s i n

a

e

o

)

.

E

.

,

e

s

s

.

h o w h w 1) t h latitude m y b accurat ly f u d fr m b erved e ith d i ta ce f a b d y f k w d c l i ati o ear t h e 8 wh e meri d ia a um i g appr xi mate val ue Ex 2 1 .

o

s

S

.

o

s

z n

n

,

ss

n

e

(

o

n

as

an

a

o

o

o

n o

e

e

n

e

n

n

o

n

o

n

n

an

R IG HT

1 14

A S C EN S I O N

AND

[

D EC LINATI O N

OH

v

.

extr mely mall d m aki g i B B i 1 w b ta i t h d e i r d r u l t d T ab le f X Y i each d y th r ugh ut t h y ar gi ve i t h ph meri d Z f 23 A u m i g t h M i l ky W y t b a g reat c i r c le f tar cutti g E 18 d m aki g a gle rthward m ea u red t h equ a t r i with t h equat r det rm i e t h d dec l f i t p le 24 A pla et h el i c tr ic rb it i i cl i ed at a m all a g l i t t h E h w th at i f i t decl i ati i a max i m um e ith er t h m ti i e c l i ptic latitude va i he t h l g itude i appr x i m atel y 90 + i t i wh re i t h l gitu de f t h a c d i g de i a m ax i m u m t h pla t P m u t b 90 fr m t h A t h d ecl i ati i ter ecti N f i t rb it with t h equat r T h pr j ecti f N P t h ecl iptic w ill al b early L t N T b t h per p dicu lar fr m N T 83 t h ecli ptic wh ere T i t h ver al q ui x de d 88 t h a ce d i g mall t ri a g le N T T w h ave t d i th I th i TTt NT tri a gle N T 88 w h ave t N T i ( PT ) t i T T) H e ce i i ( t i T T= t T T= i d appr x i ma tel y t i whe ce t h pla et l gitude i i ge eral 9o + i t i ca e

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r ve that f tar which ri t t h rth f ea t t h rat at which t h a im uth cha g i t h ame whe it r i e whe it i d ea t i d i a m i i m um wh e t h a i muth i t i — A r th f ( 2 a t wh re ) i t h latitud d t h al titude f t h tar whe d t [ M ath T r i p 190 2 ] f t h al titu d S h w th a t b ervati f tw k w tr 27 E at a k w Gr e wich ti me u ffici e t t determ i e t h latitude d l gitude f t h b erver Sh w h w fr m th e b rvati t h p iti a terre trial gl b e f th b rver m y b f u d g raphic all y tar ch e f b ervati f t h m eri d i a If t h pp ite i d e h w th at t h err r i latitude d l gitude d t t h m all rr r i t h b erved al titu de f ea h tar r pecti v l y Ex 2 6 .

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tical re fracti m ical re fracti A str Ge eral th e ry f at m ph eric r fracti I t grati f t h d i ffere tial equati f t h e refr cti atm pher ic refracti f rm u la f Ca i i atm ph er ic refr cti O th r f rm ul a f d te m p rature r fracti Eff ct f at m pheric pre u re f at m pheric re fr cti fr m b O t h de te rm i ati vati h ur a gle d decl i ati E ffect f refracti t h app re t d i t c e f t w i gh b u r E ff ct f refracti i g c le ti al p i t th a gle f a d ub l tar Effect f refracti p iti Th e

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r e fr a c t i o n t I f a ray f ligh t A O ( Fig 38 ) m ovi g through o e t ra s pare t ho m ogen eous m ediu m E H e ters at O a differe t t ran s pare t ho m oge eous m ediu m KK the ray ge e lly u dergoe a su dde cha ge f direc t io d t raverses t he n w m ediu m i t he d i tio OO T his cha ge is k ow as ef a cti o T h ra y A O is called t h e i cide t y d t he ray 0 0 t he re fracted y d both t he i cide t ray d the re fracte d ray lie i t he a m e pla e through the or mal t O t o t he surface se parati g t he m edia orm al a t O t o t he sur face separati g t he t w L t M ON be t h is k ow as the a gle f i ci de ce d m edia t he A N OA as the a gle f re f cti n d t he fu d a me tal law is e pre sed by the f r m ula o f re fractio

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the m ediu m K K is parallel to the u pper surface the ray its e m ergen ce at 0 i t o a secon d layer f the m ediu m E H o O A w hich is parallel to the i ci d e t w ill pu sue a directio directio A O T hus w e lea that a ray f light o n passi g through a pa ra llel S ide d ho m og e eous plate is o t cha ged i doubt be shi ft e d laterally A s w e are d irectio though it w ill f ra ys the lateral shi ft o w o ly co cer ed w ith t h e d i e t i o s eed o t be at t e ded to L e t u be the re f c t ive i de x fro m m ediu m E i t o H L Fig e t f be the re ractive i de fro m E i to H ( u it is r equired to fi d the re fr ctive i dex fr o m the m ediu m E i to H f

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ray fro m H through parallel plates o f H a d H e m erges i H parallel to its rigi l d irec t io ; d i f i d 6 be t he uccessive gles f i cid e ce t h e fro m t h e first i cide ce n d t h e last e m erge ce w e have the equa t io s A

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t r o n o m i c a l r e fr a c t i o n T h e rays f light from f m celestial body si g ro outer as p s pace t hrough t he earth s at mos phere u dergo what is kn o w as 40

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the u pper regio s o f the at m osphere t he d e si ty f t he air is so s mall t ha t bu t little is t here tributed to the total re fr ctio re frac t io w i t h w hich Th astro om ers h a ve t o deal takes place mai ly withi very f w m iles o f the ear t h s sur f ce I co seque ce f re fraction a y o f light fro m star does o t pass through the atm osph ere i a straight li e It follows a curve so tha t w he n the observer eceives the rays the s t ar a ppe r s t o h i m to be i a direction w hich is n o t its true directio a s tr o n o mi ca

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light co m i g towa ds us fro m a dista t star i the directio S A ( Fig 4 0) pursues a st ra igh t p th u t il it e ters the e ff ective a t m os phere a t A d from t he ce the path is n lo ger straigh t From A t o the observer t O t he ray is passi g through at mospheric layers o f w hich t he de sity is co ti ually i cr e si g so tha t t he ray curves m re d m re till i t reaches 0 To the observer t he rays a ppear t o co m e fro m T where O T is the ta ge t to the curve at 0 I f through 0 a li e OB be draw n i which the parallel to A S t his li e w ill how t he d irectio s t ar would a ppear i f there h d bee re fra cti g disturb ce T hus the e ff ec t f re fractio is t o m ove the a ppare t place f t he s t ar through t he a gle TOR p to w a ds Z the e i t h f the observer R e frac t io is greatest at the hori o where objects are a ppare tly eleva t ed by t his cause through ab ut T h obser e d coor di ates o f a heave ly bod y m ust i ge era l receive correctio s which will ho w w hat the coordi n ates would have bee had t here bee n re f ac t io Th i ves tiga t io f the e ffec t s f re frac t io is t here fore i mporta t part f pra ctical a stro o m y A

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appro i m a t e t able is here g ive showi g t he a m ou t by ces f s t ars w hich re fr a c t io n d i m i ishes t he appare t e i t h d is t d t he t her m o m e t er T h baro m eter i su pposed t o s t a d a t 30 i m y p 4 33 at 5 0 F S N e w c mb s Sphe r i c l A st An

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th e o r y o f a tm o s ph e r i c r e fr a c ti o n We shall su ppose t hat t he ear t h is spherical at m os phere is co mpose d f a successio o f t hi layer s bou d ed by spheres co ce tric w ith the ea r t h Th e re fr c t ive i d e f the air throughout each layer is to be co sta t but it m y vary f o m o e layer to a other C o sid er t w o h layers A d B ( Fig T h re fractive i d e x f t he outer layer A is relative to free ae t her d u t hat o f B is “ A ray passi g through A i the directio P Q F 41 is be t i t o t he directio QR as it passes i to B 41

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M O S P H E R I C R E F R A CT I O N

123

e pressio to be i tegrated co t ai s t w va i ables d which m us t be relate d I f the law f t his relatio w ere k w the we coul d e press i t er ms o f p so that the proble m woul d be t he i teg a tio f a cer t ai fu ctio f I B t we have t m recise i f or a t io as to t he law acco d i g to which the i de x p f re fractio varies w ith the elevatio above t he earth s sur face It iS ho w most i teresti g to fi d that it is possible to obtai a n a ppro i m ate solution O f the proble m quite s u fficie t f m ost f an r ose w i t hou t k o w ledge the law acc rdi g to which u y p p t h e de sity o f the at m os phere di m i n ishes w ith the elevatio above the earth s sur face W e shall a su m e r /a 1 s where s is a s m all qua tity because the altitude o f eve t he highest par t o f the at m osphere is s m all We shall substi t ute this i co m pa riso w ith t he ear t h s radius value f /a i n the e xpressio o f dp d disregard all powers o f W e thus have 3 above the first Th e

n

x

n

n

n

x

r

n

r

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n

'

o

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no

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ever

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.

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it

1

(W ’

o

P o si n

A,

p (p

1

P

2

p

.

.

si

’

.

Si n

i

a. s n z , o

p

(”

a

P0

1

2

2

du si n

2

Z

dp

z 2 0

ll

’

“0

°

(“

1

Z

su

p, si n

a

2 0

P

z

du

si n

°

re f ctio is thus e pressed by two i tegrals f which t he first d m ost i m porta t part e presses w ha t t h e re fractio w oul d be i f s 0 i e i f the earth s sur face was a pla e T his is f course a w ell k o w ele m e tary i t eg ral d i t s value is Th e

ra

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If

w e de n ote by

be writte

x

o

.

an

1

“

( I‘ -

o

s in z

)

the s m all qua t ity (n n

1)

o

the i tegral n

ma y

n

si n

"

1

x

) si n z }

z

this w he developed i po w ers f by M c l i t heorem w ill be co ve ie t f calcula t io If we eglect all po w ers f 1 ) ta n 1) ta z above the seco d we see tha t ( + 5 (u is t he a ppro xi m ate value f the firs t i tegral evalua t i g the seco d i tegral we are to o t ice t ha t s I e t ers as a factor i t o the i t egra d d there fore w e shall m ake

an

d

n

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n

n

x

n

or

n

n

n

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n

o

°

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n

an

n

°

AT

MO S PH E R IC

R E FRA C TI O N

a ppreciable e ror by pu tt i g u n 1 f qua t i t ies o f t he sm all t hat they m y be eglecte d T hus 1) are ord er ( t he seco d i t egral assu m e t he si m ple for m si cos [ L t m be t h e d e i t y f t hat at m os pheric shell w hich has y as i t s re frac t ive i de t he n by G lad t o e d D ale s la w u a d m are co ec t e d by a equa t io f the f r m n

r

n o

8

o

:

,

n

.

s

n

n

n

a

so

u , o

or

F0

n z 3

n s

e

n

1

.

o

.

x,

n

s

n

n

n n

z

o

’

an

n

o

cm ,

1

p

.

where c is a con s t a t qua ti t y so n

n

,

da

be t he de ity t he i t eg ral beco m es I f mo

ns

o

dm

c

.

t he air a t t he sur fa ce

f

o

t he ear t h t he n

f

,

n

s

.

dm

.

I t egratin g by parts t hi beco m es n

s

t he ter ms i n de pe n d e n t

i n tegral va i h a t bo t h li m i t ; we also make = s whe m = o d s = 0 whe m = m T h i t egral i this e pressio h a re m ark ble ig ifica ce f i t is obvi us t ha t it e pre se s t he t o t al m ass o f air lyi g ver t ically over a u it area t he ear t h s sur face d is t here fore propor ti l t o t he pressure f t he at m os phere i e t o the heigh t f t he baro m e t er T hus the actual la w by which the de sity f t he a t m osphere m y vary with t he al t i t u de is t w require d i t he proble m T h e theoretical e pressio n o f t h re fractio has there f re assu m ed a re m arkably si m ple for m It is the diff ere ce betwee two i teg als whereo f t he firs t has bee f u d d the seco d m ust be pr portio al to t a + t Fro m t his w e lea that the t o t al re fractio m ust be f t he for m A t a + B t where z is the appare t e ith dista ce d A B are certai co s t a ts T h values o f t hese co sta t s are to be deter m i ed by observatio as is shown i 46 We ca also a ssu m e various hy p theses as t o the relatio betwee r d u a d co m pare t he results so calculated with fo r

o

f t he

n

s

s

’

n

s

n

x

n

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a

as

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n

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n z

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AT

to

M O SP H E R I C R E F R AC TI O N

ake the m s t ric t ly parallel We might there fore a t ici f d m a t e t ha t w i e e arture ro tru t h will arise by as u i g d m p p t he at mosphere to be i hori o ta l layers i whi h case the o hom oge eity pro duces e ff ec t the t o t al re frac t io ec ti g t he re frac t io wi t h t he e i t h dista ce Th for m ula co i n t he case f a su pposed ho m oge eous at m os phere has bee thus ob t ai e d by C assi i We shall a su m e t ha t t h at m osphere is co de sed i to the space be t wee t he two spherical shells o f radii CS Th d C V res pec t ively at m sphere is c sidered f u i for m d e sity d o f f t ive i d ex a I t i i ges Th LI m p y t he a t m os pheric sur face t o o d which OI H is or m al re ches t he observer o t he ear t h sur face at S s tha t a gle o f z LI H = p is t h i ci de ce d [ SI C e is t he n gle f re frac t io F m 44 h T h e ray reaches t he server i t he directio I S so t hat A I S V = is t he appare t e i t h dis t a ce f the obj ec t I f de otes as be fore the radius f the ear t h a d l the t hick es o f t h at m os phere S V we have fro m t he tria gle S CI °

6

m

n

.

s

no

n

z

n

n o

n

nn

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n

,

c

n

on

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n

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n

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l /a ) s i n gb si n m f l

1 (

also p si n p H e ce l /a ) s i p (1 i very early si ce l /a is a s m all qua t i ty es t i m ated at less tha an

d

n

xr

n

1 8 00

,

(

.

.

n z,

t

n

n

n

.

be the whole re frac t io i e t h e a gle through w hich t he i ci e t ray is be t fro m i t s ori gi al directio w e have i 4; p i n d assu m i g to be e ressed seco ds f p p n

If p d n

n

n

a

ubs t itu ti g n

x

fo r

n

.

n

n

p

S

.

,

si n

si n a r ,

i

1

”

n

n

(

Sin (

n

si n « f

t

si n

i

,

co s (

5

o

«f

,

,

a re

p

se e (

.

respec t ively t he e x pressio s n

AT

p

s

1 {

(p A

tan z

l /a )

(1

c osec 1 {t a n

1)

.

l /a ) s i n

1 (

co ec 1

1)

(p

MO SP H E R I C R E F R A C TI O N

si n

2

an t (

z

z 2

fi “

z

3

ta n

2

z

) l /a }

° B t a n z +

co ec 1 1 B p ( ) l / cosec the prac t ical applica t io o f t he f r mula w hich has thus F bee d erived by t he d i ffere t pr ce ses f t his d t he precedi g ar t icle w e must obt i u m erical values f A d B This has to be do e fro m ac t ual ob erva t io f the re frac tio i t le st d we hall assu m e it has t wo par t icular i sta ces ( see bee t hus fou d t hat at t e m pera t ure 5 0 F d pressure 30 i the re fr c t io s a t the app re t e i t h dista ces 5 4 d 7 4 are res pectively d T h for m ula ( i ) will thus give fo t he d eter m i a t io f A a d B the tw o equatio s w here

(p

A

l /a )

1) ( 1

.

s

a

.

o

n

or

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n

a n

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a

.

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.

r

e

n

n

o

n

n

A A

20 0 4 6 ‘

B (t a n

(ta n (t a n

B (ta n

olvi g these equa t io s we obtai the followi g ge eral e pressi fo the re frac t io a t m ea pressure 30 i n d t e m pera t ure 5 0 F 8 2 4 8 2 t a 5 9 t 0 06 6 p tha t u le s t beco m es very T hus B /A is o ly s great i e u less t he obj ec t is ear t he hori o we m a y eglect the eco d t er m t he re fractio m y I f t h e e i t h d i t a n ce d oes t e ceed be co m puted wi t h su fficie t accuracy f m a y purposes where o e t re m e te m perat ures are i volved by the si m ple e pressio S

n

n

r

on

x

n

n

n

n

n

n

a

.

°

.

,

‘

an z

n

,

s

n

n

o

n

3

z z

a n

s

n

n

.

.

‘

z

n

z

n

,

.

z

n o

s

n

x

n

x

n

n

or

n

x

n

[0 t a n

a

n

z,

here w e are usi g on ly the first ter m that is eglecti g the ter m c t ai i g t i t is slightly m ore accurate to t ake k = rather t ha T h e qua ti ty h is ca lle d the coe fficie t o f re frac t io E 1 f a h m ge e u atm ph ere W ha t ught t b t h thi ck e which w ul d gi ve expre i f refr cti i acc rda ce wi th b

an

d w

n

n

on

an

n

3

x

.

.

o

n

z,

n

n

n

n

,

n

.

o

o

an

e

ss o n

e

n

or

B /A

ss

a

l /a ,

o

o

on

n

o

o

n

o

n

s

os

o

se r

128

AT

wh e c that taki

M O S P H E R IC

RE F

RA CT IO N

e

n

n

so

Ex

2

.

g

a

m

= 39 5 7

a

on

we fi n d l = 4 5 m

s,

i le

s

.

h w that t h refractive i dex f t h e atm ph ere w u ld be d temp 5 0 F acc rd i g t o Ca i i th e ry f at pr ure 30 i S

.

n

os

°

an

.

o

n

e

o

e ss

refr cti

i le

’

ss n

n

o

.

.

o

s

.

x

S

.

o

s

.

ns

n

S

,

n

ss

e e

.

e

n

s

s

o

n

o n

e

e

an

o

r

’

a re

or

as

e

r

’

an z

’

e

an z

o

s

e

lc = h ( 1

re

ss

o

1r

’

o

se

on

a

d b e di s

on

e o

m

itt

ed

e

s n o

o

t he

.

n

ss o n

e

at

on

°

e

ss

n

on

ze n

n

'

n

e s

x

°

n

s

.

.

s

o

o

o

ze n

n

o

.

i t h e refracti h w fr m f rm ula ( ii ) that t m p = 50 F appare t ith d i ta ce 61 48 th a t h fi fth part f a c S h w th at i f qu a titi es le 4 E regarded t h ec d term i t h expre i f t h refracti m y b whe ever t h e ith d i ta ce d e t exce d l t xpre t h refracti wh re E 5 If w th t wh e i t h pp d i ta c i t ead f i t h u ual f rm h t b th expre ed i ec d f arc d i d i ta ce h w th at if k Ex 3

o

se c

2

z

n s

2

s

Is

e

e a

on

ru e

t

a r en

e ith e ith

z n z

n

s o

i

z s n

t a e f o r a tm o s ph e r i c r e fr a c t i o n It is obvious t ha t the de si ty f t he air co s t ituti g t he a t mosphere di m i ishes as the dis t a ce fro m t he earth i creases T h e i de x o f at m ospheric re frac t io will i like m a er d i m i ish fro m its value at t he earth s sur face to t he v lue 1 a t the u pper li mits o f t he re fracti g at m osphere W e take as i 4 1 t be t he ra dius f t he lowest a t m ospheric layer f w hich p n d the ra dius o f the layer whe p has w here n is decli e d t o u ity S i mpso ass m e d that a qua t i t y at prese t u k ow Th e assu m ed equatio gives = w he p = 1 as already arra ged A s i creases is t di mi ish d this w ill be the case provided ( + 1 ) be positive W e have see 4 1 ) tha t p s in 91 co s t Equa t i g the x d lower li m i t o f t he pressio s o f t his pro duct fo the u pper a t mos here p 44

O h e r fo r m u l

.

.

o

n

n

n

n

n

n

n

n

.

nn

n

n

’

a

n

a

o

n

or

.

n

n

r

r

o

an

o,

r

’

n

n

.

n

n

’

n

n

n

n

n

.

n

r

.

n

u

,

an

,

.

u

n

o

.

u

.

r

n

n

r

p

w here a

t

n

)

o

.

an

=r

a srn z

I

s rn z

n

o

n

.

p

w he n ce

oa

sin z si n z

or

r

n

a no

F

r

” o

’

“

si n z

si n z

’ ,

s

,

n

n

e

I

is the a gle f i cid e ce at the u pper mos t the lowes t S ubs t i t uti g f we have z

’

n

’

,

an

d

z

AT

M O S P H ER IC R E F R ACTI O N

or as t he re frac t io is sm all n

,

,

n

+

1

5 n p)

(z

tan

.

we i t ro duce the values f p d use d i s h oul d fi d as t he a ppro i m a t e for m ula If

o

n

an

,

u

.

.

,

,

x

n

”

(z

59 tan

p

4 p)

.

We correct t his for m ula so as t o make it e act f k ow re frac t io s a t t a d ard te m perature d pressure e a m ple we t ake x

ca n

two

or

an

n

s

n

n

n

we

Ex 2 p 1 29

n

If

,

fo r

x

z

69 36 ‘

p

d

an

z

2 14 1 0

p

f m see ree wich T ables we ge t t he fi al r G ) ( n

n

,

ta n

p

o

(z

4 0 9p) '

.

t his for m ula all re frac t io n s u p to the z e n i t h dis t a n ce o f 8 0 c a n be de t er mi n ed a ppro x i m at ely B ra d ley s for m ulae is s ui t ed fo r observa t io n s n ear t he hori z o n °

By

.

’

because t approaches

(

an

Ex

C

1

.

a i i ss n

v iz

,

z

90

S

.

4 09p) d °

oes

beco m e i d efi itel y large as

n ot

n

,

z

n

°

h w that

r ula

fo m

the

o

r fracti

fo r

e

on

g

i ve

by B

n

rad ley

an

d

.

= 58 p

and

‘

3 61 t a n

tan

p

(z

4 09p) '

-

ta n 3 2 ,

z

r ctically qui vale t u ti l t h e ith d i ta ce b ec me very large E 2 upp iti that t h ( + 1)t h p wer f t h i dex f O th refracti f th atm ph er var i i v r ly th d i ta c e fr m t h ce tre f t h earth pr ve B rad l y appr xi mate f rm ula f a tr m ical r fr cti p = t ( —5 p) O x f rd Se i r S ch lar hi p 190 3 E 3 I f i t h a tm phere t h i d ex f re fracti vary i ver ely t h arth ce tre b i g p at t h earth urface quare f t h d i ta ce fr m t h d u ity a t t h l i m it h w that t h c rre p d i g f t h at m ph re c rrecti f refr cti i g i e b y i ( + 5p) = J p i Math m atic l T ri p 1906 a re

p

a

e

x

on

e

o

o

n

a

.

s

a

an

e

n

or

e

45 5 r e fr a c

E ff e c

ti o n

t

o

f

a

e

s

tm o

as

se

v

n

s

os

o

e

e

o

n

e n

,

S

,

o

s

s

on

.

as

’

e

o

on o

s

e

o

e

,

n

r

o

n

n

s

or

n o

o

e

.

o

o

s

’

e e

S n

"

e

’

e

o

on

n

s

o

o

o

a

o

n

n

.

os

e

on

es

e

n

e

s

n

o

2

n

.

e

o

,

s

on

os

os

e

o

an

s

e

on

x

e

n

.

.

e z n

n

n

o

e

s s

s

on

n

n

z

0 s n 2

s ph e r i c

pr e s s

a

e

.

ure

an

d

o s,

.

t e m pe r a tu r e

on

.

or mula ( ii ) f the re frac t io already obtai ed 4 3) w assu me d t ha t t he baro m eter s t o d a t 30 i ches d t he e t er al air at the " te mpera t ure 5 0 F We have w to fi d t he f r m ula w t be used whe ressur te m era t ure have other k o e d p y p values I n t he f

or

n

n

e

.

o

°

.

n

n o

an

n

n

x o

‘

o

~

.

n

an

an

n

n

4 5 — 4 6]

AT

M O S P H E R IC

R E F R AQ TI O N

W e assu m e t hat t he re frac t io n i s propor t i o n al t o t he de n s i t y o f the air a t the e a r t h s surface so t hat i f p be t he re fra c tion fo r m t an d f n h e re t e era t ure t he re ractio a t and t ss ure p p p p s t an dard pre s sure 30 i n ches a n d t e mperature we ob t ai n fro m t h e pro per t ies o f g a s es ’

,

0

1 719

4 0 0 5 6 + p

p

I t r duci g the value f p already fou d 4 2 ) w e b t ai t he a ppro i m ate f rm ula f a t m o pheric re fra t i at pressure p t h a ppare t e ith d i t a e d t e m perature t f n

o

n

o

P

n

or

e

l 7p 4 60 + t

8 5 (

an

°

t he a ppe n di x to t he

In

ta

2 94

c

n

z

n

o

s

or

o

x

n

0

s

on

nc

z

.

0

n z

.

Gr e en wi ch Obse r v a ti o n s

f

1 8 98

or

Mr

,

owell has arra ged tabl es f re fra t io which are use d i T hese t ables c t ai t he m ea re frac t io G ree w i h observa t ory every m i u t e d t e m pera t ure 5 0 F f t he pressure 30 i he f Th orrecti s whi h f e i t h d i t a ce fro m 0 to 8 8 d pressure are give m ust be a pplie d f c h a ges i t e m pera t ure i a dd i t io al t ables P H C n

c

nc

z

n

n

n

46

n

n

or

.

°

on

c

an

n

n

n

c

e

n s

n

.

O n th

.

n

°

°

n

s

n

s an

or

o

on

.

or

o

c

o

n

.

.

t

b se rv a i o n

ete r m i

d

e

ti o

n a

n

o

f

a

tm

o s ph e r i c r e

t

fr a c i o n fr o m

.

We d escribe t hree f t he m e t h ds by whi h t he oe ff cie t t he e pre sio f the re frac t io A t A d B i +B t be d e t er m i e d by observatio f m eri dia e i t h d is t a ce T h firs t be carrie d ou t at a si gle d m etho d s d se observa t ory pr vi de d i t s la t i t u de is ei t her very great very m all f two b T h t hir d m e t h d requires t h coopera t io t i i t h s u t her h e i ph i t he or t her do f F i r t M e th o d elec t e d such t ha t i t will be ab ve A s t ar i t h h ri b t h a t u pper d a t lower cul mi atio I f be th appare t e i t h dis t a ces a t lower d u pper cul m i atio respec t ively or t h f t he e i t h t h e d p si t ive t o t he t he t rue e i t h d is t a ces will be + A t a + B t d A t e i t h dista ces B t Th m ea f t he e t w o

o

an

s

x

n

ca n

or

n

n

n

n

e

o

va

o r e s, o n e

o

n

zo n

z

z

’

o

an

z

i i

ri

o ds o f g me b e d b y Si r D a v d G

l S o ci e t y , V o l

.

v

1

.

i

p 32 5 .

n

O b se

ill

.

o

rn

s

se r

e re

n

z, z

.

an

e

th

r e ma n n

f L o e wy d e sc

n

.

ri

v n

i n th e

o

z

.

n z

s

r r cti

’

n

o

z

.

o

n

o

n

’

.

o

n

3

z

s

s

n

n

°

n

n

n

e

n

n e

o

an z

n o mi c a

s

n

'

T Of th e

an

s

n or

an

an

z

n an

n

o

n

z

e

.

e

n

o

s

e

an z

,

n

e

.

n

i

n

o

s

c

ca n

co n

an

c

o

n

n

,

an

z

n

°

z

n

an

n

i

th t

o n s we m a y m e n t o n a g ef a M o n t h l y N o t i c e s o f t h e R o y a l A s tr o

AT

.

e

[

o

f

n

,

}{

z

z

1

’

A (ta n

'

ta n

z

n

o

z

B

)

o

n

n

n

.

.

z

v1

ca

course t he dista ce fro m t he e i t h t t he or t h t he cola t i t u de H e ce we obtai t he equatio

i s, i

M O S P H E R I C R E F R ACTI O N

p

le

,

n

(t a n

3

tan

z

90

3

°

ubs titu t i g t he observe d values f nd we ob t ai a li ear equati be t wee the three qua ti t ies A B d d each star O t her s t ars are also observe d i t he a m e way gives equa t io i t he sam e t hree u k ow s T h ree f such equa t io s will su ffice to d e t er m i e A B T h e resul t will h w ever be m uch more accura t e i f we Ob erve m a y star d t he t rea t the resul t i g equa t io s by t he m e t ho d f leas t quares t o be subseque tly d e cribe d w e shall t ake a case i whi h t he A s a si m ple illus t ra t io la titu de is k o w d i which as ei t her f t he e ith d ista es is e cessive we m y assu m e t hat t he re frac t io is e pre se d by t he si gle t er m I t A t D u si k i N lati t u d e 5 3 2 3 1 3 t h e star C e phei is h serve d t o have the appare t e ith dis t a ce 8 4 8 37 at u pper cul m i a t io A t l w e cul mi a t io 1 2 hours la t er i t s a ppare t e ith dista ce is 64 2 2 T h e true e i t h d i t a ces will be S

o

n

n

on

z a

n

n

s

n

n

,

n

n

n

n

s

s an

o

n

x

.

an

n

n

n

n

n

z

°

n

8

4 8 37

k

"

su

m

t he s e

f

o

w hen ce

'

m

s

r

o

m

’

°

’

8 48

( 64

2 2 47 °

( 36

o

7 3 13

[c ( 0 1 5 5

73 11 2 4 f

°

u t be d uble the cola t i t u d e

’

°

n

ta n

[0 t a n

64 2 2 4 7

Th e

n

n

’

°

”

’

s

°

n

o

°

n

n

z

r

n

s

a

.

o

z

nc

n

x

’

°

n

.

n

z

.

n

n

o

n

,

n

an z

t

c

n

a

,

n

s

n

n

o

,

s

n

o

.

,

,

n

n

an

n

n

n

a n

,

n

an

z

’

which

°

’

k

th o d T h e co s t a ts f re fractio also be f t he ls t itial e i t h dista ces f d e t er m i e d by observatio the be t he a ppare t meri di al e i t h d ist ce f t he L t a t the solstices L t p d p be t he corre po di g re fractio s A m T he the true e i t h d is t a ce are d ssu i g p p that the su s la t i t u d e m y be eglecte d or i o t her words tha t t he su s ce tre is actually i t he ecli ptic as is al w ays very early true w e obtai f the m ea f these e i t h dista ce the arc fro m the e ith t o the equa t or t the latitu de H e ce w e have Se c o n d M

e

n

.

n

su n

z

ca n

n

n

o

.

z 1 , z2

e

n

so

o

n

o

n

o n

n

su n

e

.

n

z

n

1

an

n

n

’

z

n

s

,

s

1 an

z1

n

a

an

n

z2

s

n

,

n

n

.

n

,

o

,

’

n

n

n

n

n

,

z

or

n

n

z

o

,

= zi 2< > l

.

e

.

.

P1 + P2

n

n

°

s

n

.

AT

M O SP H E R IC

RE

FRA CTIO N

[

OH

VI

.

e i t h di ta ce is A where A = k t We a su m e tha t w t ha t A g! 0 d as t he a i m u t h d es t t h latitu d e i k al t er by re frac tio A 0 d e li atio we w ri t e t he f r m ula T fi d t he e ffec t

z

an z

z

z

z

n

s

n

s

.

,

s

e

n

n

n

on

i g A d , A d) A z

n e ct n

,

AS

co s

an

,

z

o

n o

.

n

o

)

so

no

AS

,

n

c

n

o

co n

35

,

n Az

co s

sin

b A cp

h

cos ¢ A

a

0,

which with t he sub t i t u t io A 0 A 4 0 A k t gives cos 7 t e i f 8 i t he bserve d d ecli a t io t he k ta AS 8 [ t 7 is t he true d e c li a t io hour a gle we have 35 T fi d t he e ff ec t n

s

n z

1

on

Az

co s a

z

,

o

s

.

n

n

n

o

;

,

a n z co s 7

0

f

.

,

a

an z

n

n

n

.

n

A c[>

co s

nAB

si n a

co s

Ah

0,

ro m w hich by the sa me substi t utio s n

sec 3 aralla t ic a gle we use p k s i n 7; t a n

Ah

Fo r

the e ffec t

on

cos

A 77

d fin d

an

.

SA IL

si n

si n a s i n

k s i n 77 t a n 8 t a n

A 7;

35

n

c

z Aa

z

z

q

5

0,

.

resul t s just obtai ed m y be o t herwise prove d as follows Fig 4 6 N is t he N or t h P le Z t h e i t h P t he t rue place f t he star as raised the star d P t he a ppare t place Th e

In

n

a

o

.

of

e

,

’

an

,

.

z

n

F IG

t owar ds t he z e n ith by re frac t i o n

,

o

46

.

n

.

d P P = k t a n ZP = k t a n ’

'

,

an

z

.

er pe dicular to P N d p i d d t h 4 P N P is s m all as will be t he case u less P is ear t he pole t he ha ge i polar d is t a e is P P cos ; cos PQ [ t T h observe d d e li a t io is 90 N Q b t t he real d ecli a t io i — 0 NP H e ce t h e observe d d ecli atio is t oo large 9 d

P

'

Q

'

is p

ro v

an

n

e

e

n

n

’

,

c

,

n

n

n c

’

7

e

c

n

n

0

an z

°

u

,

n

n

°

.

n

n

n

,

an

s

4 7— 4 8 ]

AT

M O S P H E R IC R E F R A CT I O N

co seque t ly t he correc t io A 8 t be a pplie d t o t he bserve d b t ai t he t rue d ecli a t io i give by d ecli a t io t — kt A8 cos 7 We have also i 7 cose c P N kt kt Ah P NQ i 7 sec 8 A i 7 cos 8 is u altere d by re frac t io we m us t have 8 i i cos 8 A 8 A 7 7 7 whe ce by substitu t i g f A 8 we fi d n

n

n

o

n

n

o

o

n

n

o

n

an z

’

7

an z s n 7

.

n

n

cos 7

n

.

’

an z s n 7

s s n 7

n

s

s n 7

7

,

n

or

n

s n

k si n 77 t a n 8 t a n

A77

z

.

t o f r e fr a c t i o n o t h e a ppa r e n t d i t a n c e b e tw e e n ti a l p o i n t t w o n e i gh b o u r i n g c e l We shall firs t sho w that i f the re frac t io be take as k t t he the c r e ti t o be a dd e d t o the a ppare t dista ce D i seco ds f e be t wee t w o eighbouri g s t ars is i seco ds f arc kD ( 1 cos 6 t ) i l w here is t he e ith d is t ce f the pri cipal star d 6 is t he a gle betwee t he arc j oi i g t he t w o s t ars d t he e fro m the ri ci al star t o t he e i t h p p L t Z be the e i t h Z A = w Z B = y A B = D é A ZB T h e ff ec t f ZA B 0 is t o m ove the arc frac t io A B where t AB p 48

E ff e c

.

n

s

es

s

.

n

n

n

o

n

r

c

on

n

n

ar

o

n

n

2

z

z

an

n

n

n

n

n

z

e

n

o

”

s n

z

n

,

n

an

n

an

ar

.

a

z

,

,

,

,

,

re

o

e

.

an

n

n

o

n

z

n

2

a n z,

n

’

o

u

AA an

BB

d T

he

’

’

k

’

[6 t a n y

tan w .

F IG

.

n

cos D cos cos y D i ff ere tiati g w i t h as co sta t n

n

a,

n

fin d

— si n D

.

n

,

an

cos a

k

ta n

a:

Ay

h

t an

y

co s

y tan

co s x si n

a

y

.

n

Ax

A D = k si n w c o s y t a n w + k

10

cos d m aki g

si n no s i n

a:

we

47

.

z

,

w si n —

k

y ta

eos

n

!

3

a si n

w co s y

tan

y

sec y i y %a i the e i t h A both t hese t er m are m all we m y p t y tar i t he e pressio s sec sec y d ista ce f ei t her d i i y A lso si ce D are s m all we m y p t = i D=D d i ( s 9 y) D k s i n ( a:

y)

2

n

s n ac s n

s

o

n

.

s n

as s e e

s

s

s

,

a

2 i 4k s n

a

n

z,

as

u

x

n

a:

a

.

an

s n a: s n

s n

2

x

2

co

u

2

‘

.

z

n

an

AT M O S P

also

4 si n

we t h us obtai

d

an

at

2

D

2

a

kD ( l

or i f k

si n

2

t he d e c rease

fo r

n

H E R IC R EF R A CTIO N

co s

2

2

in

[

c ec D d to re frac t io 6

?

os

0 tan

.

VI

z

u e

2

OH

n

)

z

are e presse d i eco d s f [ D (1 cos 9 t ) i 1 give t he ec ds by w h i h D h bee lesse e d by re frac t io ; t his is c seque t ly the orrec t i t t he m easure d d ista ce be t wee t wo eighb uri g stars to clear fr m the e ffect f re frac t io We have e t t o show t ha t 9 t he a gle which t he li e j i i g the t w star m akes wi t h t he ver t i al i i crea e d by re frac t i to t he exte t k i 0 (9 t T aki g the logari t h m i c d i ff ere tial f t he equatio ,

D, AD

s

n

x

2

C

c

on

s

s

o

z

n

n

n

co s

an

z

n

s

,

on

s

.

D Mn 6 = §

n

o

n

n

Mn y

n a

AD /D w e have co t HA G cot y Ay w hich beco m es by sub s ti t u t ion

cos

.

o n n

n

n

c

s n

n

o

,

2

n

n

s

n

s n

o

x

n

2

o

o n

n

o

n

an

a rc

o

as

c

n

on

n

,

cot GA G k whe e A0 k i 9 9 ta ub t racted f o m t h e a d this is t he q ua tity which m u t be appare t a gle B A Z t o g t t he t rue a gle B A Z T h e d e for m a t io or m oo f the ircular dis f the by re frac t io is b t ai e d follows L t S ( Fig 4 8 ) be the su s ce t re a i t s ra diu P a poi t o its li m b d Z t he e i t h d let Z S = L e t k be t h coe fficie t f re frac t io which d is places P t o P d le t P Q d P Q be d Z er e iculars Fro m wha t we have S p p j ust see n P Q is d isplace d by re fra t io t P Q I f we t a ke S as origin S Z as a is f d d y t he coord i a t es f P the k (l

0 ta n

2

n c

2

z

)

s n

n

,

co s

n

n

’

n

n

2

z,

’

r

s

s

e

n

.

'

n

n

c

o

n

o

n

su n

o

c

as

’

e

n

.

,

an

n

z

e

n

n

an

,

z

o

n

an

an

.

,

a: a n

’

’

on

.

n

'

,

n

s,

,

n

’

o

’

c

n

o

x

o

x, a n

'

.

n

,

—

k) s in 0

.

A

ls

w=

SQ = a

co s

9+

a co s

d + k tan

a co s

0+k(

QQ — z (

ta n z

'

a

—a

by eli mi a t i g 9 we have f t he re frac t e d figure f t he

an o

o

’

d

n

n

o

co

s

co s

fo r

0) 9

fi

se c z

)

,

t he equa t io n

su n

FIG

.

48

.

n

AT

k

(1

a

1r

cos

2

at right a gl n

i

e

.

at

.

6

ec n

an

R A CTI O N

arith m tic

t he

e

RE F

me

e

d 6

is

n

It

(l

a

a

o

rad ii mea u r d = ) 5D wh c

f t he

s

2 i t ta n 2 g

e

en

,

e

2 k a n h t (1 + + §

=D

2a

H

6 ta n 2

e s,

M O SP H E RI C

ect o f r e fr a ct i o t h e m e a u r e m e t o f th e ta p o i t i n a g le o f a d o u b l e L t A B be re pec t ively the pri i pal s t ar d t he sec d ary s t ar d let P be t he r t h pole f t he pair which for m t he d ouble s t ar I magi e a circle wi t h e t re A t he cele t ial s phere d gradua t e d t ha t the bserver is t h ole d t ha t A P T h p i t i which A B m ee t s t he gra d ua t e d cu t t he circle at circle is sai d to be t he p si ti B w i th e p c t t A g l e of th e t T h m o d e i whi h t he p siti a gle is m easure d m y be further illus t rated as f llows S u ppose t he d ouble s t ar is or ear t he m eri d ia d at its u pper c l m i a t io d t he sec d ary s t ar is d east f the pri ci pal star T he t he posi t io a g le is abou t If however the ec d ary s t ar h d bee d wes t whe t h e t h e m eri d ia its posi t io a gle woul d be pri ci pal s t ar w ab u t 2 7 f i each case the d irec t io f m easure m e t fr m t he t t he p le i t h e a m e A s t ro o m ers ge erally d raw k ow this as the d irecti f t he m easure m e t pr cee d s fr m the th poi t towar d t he par t f the sky w hich is f l l o w i g fro m t he d iur al m ove m e t r u d by t he d the back t o th t he o th by t he p ce di g par t f t he sky I f P be t he pole Z t he e i t h d A t he pri ci pal t ar f t h d ouble A B ( Fig the the po i t io a gle as w e have j us t d efi ed i t is A P A B T h re frac t io ha ge t he posi t i a gle i t o P A B T hus the re fractio cha ges the p itio a gle i t wo ways firs t by al t eri g t he parallac t ic a gle P A Z 7 d seco dly by alteri g B A Z B oth these a gles are al t ere d by re fractio d th correc t i t o a pply t o b erve d a gle i t he ase re prese t e d i po i tio t h fi gure m us t be eg a tive We d e o t e th t rue posi t io a gle by p F 4 9 — = We have A B A Z p 7 d he ce *

49

E ff

.

n

o

s

on

e

n

s

r

s

,

s

n

.

n c

an

on

an

o

n

c

so

e

e

o n

on

o

e

n

c

o

o

n

u e

n

or

n

n

o

s

n

s

r

re

o

n

'

c

n

,

os

s

on

n

n

n

n

n

,

n

n

7

n

.

n

n

e

on

an

n

n

c

e

n

e

n

48 > A A

o

,

an

s

n

n

n

n

.

.

7

s

s

.

n

n

,

an

IG

n

cos ( p —

B A Z = p — 77 + k si n ( p — 77) P A Z = 77 + k t a n z t a n 8 s i n 77 ’

’

n

n

n

n

n

n

an

an

s

.

s

o

'

n

an

o

.

n

.

n

n

sou

n

z

o

n

n

o

.

n

o

.

n

n

,

e

n

or

,

n

o

n

.

n

n

s

n

n or

o

u e

,

n

o

n

n

a

o n

e

n

n

on

e

on

an

,

n

n

s

a

n

n

n

r

n

o n

as

o

an

s ar

.

s

,

an

on

n

a re

n

a n

u

o

s

.

o

.

n

on

an

,

o n

n

o

s

n o

'

.

)

77

ta a

'

’

,

o

4 9]

AT

M O SP H E R I C RE F RA CTIO N

there fore p be the posi t io a gle as a ff ec t e d by re frac t io — — = cos t a t 8 i i k t k + 7 7 7 ) ( ( ) p p p p d p be t he corres po di g qua t i t ies w ith res pec t t o If p a other s t ar with re fere ce t t he sa m e pri m ary 8 i k i ( p 7) kt t 7) t 7 (p S ub t rac t i g w ea ily fi d If

an z

r

’

s n 7

an

s n

’

an

,

n

—

’

p

=

p

pr

—

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s

e

.

e

on

ss u

e

e

n

s

e

.

s

n o

s

e

e

o n

o

s

o

r

e

n

on s o

o

on o

no

a re

n

e s

e

se

o

o n

e

n

O

e on

s

n

o

e

no

so

o

ss

e

s

no

,

s.

e

o

e o

.

.

E

.

o

e

n

o

1 6d 1 8 h 5 m

Fi d while t h fifth at ll it h a peri d f 0 1 1 5 7 f th e e t w atell it fr m K e pl r thi rd l w t h rati f t h m ea d i ta c e fr m t h pr i m ary tell ite f Mar re lve E 14 A u m i g th at D ei m d Ph b s t h i cir cular rb it 1 90 9 t h b rved d th a t at t h f S ept pp iti h w fr m fr m t h ce t re f Mar i 1 g rea te t d i ta ce f D ei m i K epler thi rd law th at t h g r a t t apparen t d i ta c f Ph b it be i g g i ve th at t h per i d ic ti me f Ph b i 7 39 13 8 5 d f e

’

e

o

*

x

.

.

ss

s

s

n

e O

os

o

s

n

os

o

e

n

im

s

n

os an

n

s an

o

De

e

o o

e

m

h

d

o

o

as

n

o s

s

s o

n

es

.

n

’

e

a

s

e

o

e

s

30b

e

o

os

on o

n

e

e

s

o

es

s

o

o

o

os

'

s

n

e

“

s

s

o

o

00

se

e O

,

.

vo

s,

s o

e sa

o ,

o

os

o

s

8

an

o

5 48 8 6

.

t o f th e Su n Th rev lutio o f the ear t h ab u t the s causes cha ges both as see d the a ppare t i e f t he i n the a ppare t place We have w t o sh w t ha t the phe m e a wi t h fr m t he earth whi h w e are co cer ed i t his cha pter w oul d be e actly e 51

t

A pp a r e n

.

e

o

mo ion

n

an

c

n

n o

.

n

n

n

n

u n

o

n

o

.

o

s z

o

su n

no

n

n

x

r

K E PL E R S ’

’

[

N E W TO N s L A W S

A ND

OH

.

VII

t o f e l li p t i c m o t i o n ce tre d OP O the elli pse with L t F be t he ear t h w h ich t h F as focus i appears to m ake i t s a ual rev l u t io 0 0 is t he m ajor a is f t he elli pse d 0 its ce t re Th e circle O Q O has its ce t re at C d its r dius Th e li e 00 $0 0 a QP H is per pe dicular t 0 0 52

Calcu la i o n

.

.

’

,

su n

e

n

,

an

n

s

e

’

nn

’

n

o

.

an

o

x

’

n

.

an

n

a

n

.

n

an

d PP

=

r

’

o

L e t A OFP

.

v,

hus v are t he polar coordi ates o f P with res pect t o the origi T h e a gles F d a is F 0 F m 55 are called res pectively d v the t e o m ly d t he e cce t i c a o m l y d 0 be i g t he e tre m i t ies f the m aj or a is T h poi ts 0 h s s o f orbit hat o f t h e elli pse are t er m e d the e t T p pse 0 which is eare t t h e earth is t er m ed the pe i g e e Th e o t her apse 0 is called the p g e T h e ti m e is to be m easure d fro m t hat m o m e n t k ow as t he ep o ch at w hich the s passes through t he perigee 0 I f we had bee co sideri g the t rue f the ear t h rou d t he s t he t he p i ts 0 d 0 m o t io woul d h ave bee n t er m ed t he pe i h li o d the phe li o p ly We shoul d also ote that CE e CO e ti We have o w t o sho w h o w the polar coordi a t es o f the s are to be fou d w he t he t i m e is give It is t i deed possible t o O b t ai fi i t e values f d v i t er m s f t We ho w ever with t he hel p f the ecce t ric a o m aly btai e pres io s d v t be calcula t e d to i series w hich e able the values O f y d esired a ppro i m a t ion Fr m Ke pler s seco d law we see t ha t i f t be t he ti m e i w hich t he d i f T be t he perio d ic t i m e f t he m oves fro m O t P rbi t area OF P area f elli pse T t I t roduci g to sig i fy t he me a m o ti o i e t he circula m easure f t he average value f the a gle s w e pt over by t he ra d ius vec t or an

d

00Q = u

£

T

.

r

,

n

n

x

an

n

.

,

.

u

an

an

r u

an

a

n

e

n

’

an

r

n

a

x

n

o

a

n

.

x

e

a

.

s

r

.

’

a

n

o

e

.

n

u n

,

n

.

n

o

u n

n

r

ve

n

n

,

an

n

e

a

a

n

n

n

n

or

n

an

u

r an

can

n

O

x

,

an

O

,

o

n

n

o

n

n

,

.

.

.

n

an

n

o

n

s

o

n

su n

n

,

.

o

o

n

.

,

ec

u n

o

n

n

’

n

re s

.

n o

.

n

n

x

o

r

o

,

n

n

n

n

’

an

o n

n

.

n

r

§

5 2]

AND

T

H EI R

A P P L I CA T I O N

the u it o f t i m e we have = 2 /T d a s t he area f t he ellipse is a b w e have t = 2 area OF P /a b o m ly Th e a gle t is f m uch i m por t a ce ; i t is calle d the m e d is usually de ote d by m Fro m t he pro per t ies f the elli pse P H QH b/ whe ce i cos ) area OH P b GH Q/ b ( O CQ H CQ)/ a b (

in

n

u

,

7r

o

an

,

n

n

n

.

n

an

an an

n

o

n

a

.

a

o

a

a

.

n

,

u

, a

s n it

u

e si are F H P b QH F H /2 J; a b ( s i cos ) w he ce P HP OH P OF P } a b (u e i ) m e sin a d fi ally d we e press T hus m is e pressed i n ter m s O f ter m s o f u as follo w s Fro m the elli pse w e see a t o ce —ae c s v = a cos a

it

n

a

.

n

n

it

.

,

,

u

u

.

n u

s n u

1

n

,

u

x

v

x

an

,

:

n

a

o

r

b si n

r si n v

w he n ce squari n g ,

an

d

2r

si n

§

v

=

r

n

=

,

a

(l

cos v)

(1

,

addi g w e obtai r

2

u

,

a

n

— e co s u

(1

)

—

2r

cos

2

%

v

=r

(l

+

cos v)

a

(1

e

d

fi ally

+

u

— a an

(l +

a

cos

cos

—

e co s u

e

cos

+

u

e

)

— 1 (

)

u

e

co s u

)

,

)

(l

n

O

O

O

O

O

O

O

O

O

O

O

O

O

O O

O

O

O

H EO RE M I f we coul d eli mi ate fro m ( i ) d ( iii ) we shoul d have the relatio bet w ee m d bu t o w i g to the tra sce de tal ature f the equation s such eli m i ati i fi i t e ter m s is i m possible W i t h t he hel p o f L agra ge s t heore m w e m y however e press v i ter m s f m by a series asce di g i powers f which fo give values O f m a d e w ill e able us t c m pute v wi t h d egree f accuracy required y L agra ge s theore m m y be t hu s t a t e d — I f we are give *

A P P LI A T I N F L C O O A C RANG E [

it

’

s T

n

.

an

n

an

n

n

n

n

on

n

n

n

n

an

v,

O

n

“

’

n

a

n

n

n

o

n

,

o

o

,

x

n

r

e

an

o

n

n

o

.

’

n

a

s

n

K EPLER S ’

which fu c t io o f

in

an

93

2

n

n

d y

NE WT O N

AND

’

are i d e pe de t variables n

n

n

[

S L AWS

d if F

an

,

OH

.

be

(z )

V II

an

t he n

,

y

j in

which

di

as usual d e otes

—

n

v

apply this t o t he case be fore us w e see that i f w e wri t e f m f i m f we ake equatio e f d s i 4 ( ) ( ) y I f fur t her w e w rite ( iii ) i is ide tical wi t h equa t io the f r m v = F ( ) the w e have fro m equa t io ( A ) To

,

o r as,

or z,

u

or

an

,

n

n

,

n

u

o

F

v

n u

u

;

(u )

n

n

= E

a

n

( m)

e si n

mF

3 d l

i

si {

2 d m |

z

n

2

!

+

d

6

( m) +

’

n

2

{S i n

e tc

3

.

_

ro m equa t io ( iii ) w e fi d by a w ell k ow m etrical e pa sio which is proved O p 1 60 Bu t f

x

where

n

c

an

d

{

2

u

u

1 {

N/ I

E ( m)

n

m

0

e

2

si n

fl/

e

{

h

e

u

H

.

sin

.

2u

t rigon o

n

,

§

c si n

3u

3

e ce n

$0 s i n

m

0 sin

?

n

-

n

( )

E

v

n

n

si n

2m

2

3m

t here fore

cos m 0 cos 2 m + c cos 3m t he right ha d si de f equa t io ( B ) m y be H e ce all ter m s evaluated d t hu m y be b t ai e d w ith y required degree f accuracy S e for m ula ( vii) p KE PL E R S PRO B LE M T e ff ect t he solu t io o f equa t io ( i ) i e t o d e t er m i e whe m is k ow n is ft e called Ke pler s proble m is a ppr i m a t e value f w hich has bee S u ppose arrive d at by esti ma t io or other w ise d le t ’

F (m)

1

{0

2

n

2

on

,

an

o

a

,

’

it

n

n

u 0

an

ox

a

an

.

n

o

.

n

n

n

O

e

O

n

-

s v

.

3

o

n

,

,

.

.

'

.

o

n

n

u

n

,

an

e si n n o

t he t rue value O f we have a ppro xi m a t ely If

u

be

u 0

Au

o,

t he n by subs t i t u t ion 0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

in

i( )

K E PL E R S ’

15 8

’

Fro m the esse tial proper ty t hat t he arc A U is equal t o UT but

to FT

n

.

[ OH

N E W T O N S LA W S

A ND

f

o

.

V II

i volute it follo w s

an

n

,

UT

w he n ce

which

F CU=

a e si n

we

if

,

m

CF si n F O U =

:

ake

4

FCU

(4

a

F0 U

u

m=

,

rawi g perpe diculars UR we have

e si n u

on

u

F V co s A M E V = C U co s u

F O U, ’

A OF )

ly

mp

Si

.

UR fro m U

n

n

A

beco m es

'

D

a e sin

CF

on

CF

cos

a

:

an

—

u

r

QV

d

f

om

Q

a o,

— b i si d t he i volu t e A TJ have bee T h us whe t he t hree circles draw t he solutio f Ke pler s proble m m y be su mm ari e d as follows T ake a poi t M o t he m aj or circle so tha t A A CM = m f GM wi t h t he focal circle d ra w Fro m F t he i t ersec t i the ta ge t F T t o the i volu t e d through 0 d raw OQ U parallel d mi or ircles i U d Q t F T cu tt i g t he m aj or respec t ively T he 4 A OM = m ; é M F V = v ; £ M C U = ; F V = d t he proble m is solve d hi g such as t h se f B l greatly facilitate the solutio o f t he proble m f fi di g w he m d e are give W e illus t ra t e their use i t he follo w i g questio B ei g give t he f llowi g assu m e d ele m e t s f t he orbit f H alley s C m et E cce t ricity e 0 9 61 7 33 T i m e f P erihelio P a sage 1 91 0 M y 2 4 P eriod P 7 6 08 5 year fi d t he ec e tric d t rue a om alies f the co m et 1 9 00

F V s i n L M F V = CQ

n u

an

n

n

n

o

z

a

.

on

o

,

n

n

'

an

,

an

n

,

n

n

n

o

.

n

’

n

n

s n u

c

n

n

an

.

n

u

an

r

,

.

o

o

n

n

a u sc

o

n

n

n

‘

an

n

u

n

n

n

n

er

‘

n

n

o

n

.

.

or

o

’

o

,

n

o

n

,

,

s

a

,

s,

,

n

-

Ma y 2 4

c

n

,

an

,

n

on

o

,

.

We have t he m ea m o t io equal to to periheli is 10 years we have n

3 GO /P , °

n

an

d

i ce the ti m e

S n

on

=

10

x

3 60

x

60

x

60

7 60 8 5 l A s tr o n o mi c a l T a b l e s b y B a u

‘ ‘

sch i n

1 7 0335 s = 4 7

u li h

ge r , p b

s

ed

18

l

s

.

i i

b y E n g e m a n n , L e pz g

.

AND

eri g B s argu m e t s m o f t he ecce tric a En t

au

n

chin

n

n o

m

T

e r s g

d

fin

u ,

log cosec 1 log e i s n u ,

e si n u

9 9 9 1 4 98 4 99 8 30 5 4 7 5 3 14 4 2 5 1

Lo g

5 2 8 8 97 8 2

1

, °

2 54 1 0 1 18

v,

m

—

'

m,

182

co s n

9 2 92 1 4 9 9 8 305

,

e c

s u ,

)

u

,

2 2 600 7 0 0 7 4 96 21 85 11

15 3 15 2 33 1 5 0 = 1 0 1 18

Au ,

'

0

-

,

u ,

z

his m ust be very early the true value o f u proceed t o a seco d a ppro i m ation T

n

n

'

°

u

n

11 8 8 4

,

m, )

o log A

lo g ( l

'

as follows

Au ,

log ( m

6 2 0 0 ‘

'

m

x

e co s u

47 15 5 3 8 4 7 18 5 5 8

m, =

n

,

hen fro m for m ula ( iv ) we calcu l a t e

L o g si n Lo g 6

ouble e t ry wi t h the d the a ppro i m a t e value d

Of

aly u

T

159

A P P LIC ATI O N

ables e = 0 9 6 we

’

an

n

HE IR

T

.

~

10 1 2 0

To

33 1 5

veri fy it w e

x

Lo g Sin Lo g e

u

9 9 9 14 338 9 9 8 30 5 4 7 5 3 14 4 2 5 1

,

log cosec 1

lo g

e Si n u

,

e si n u

,

54

m

’

°

u ,

1 10 1 2 0 33 1 5

m, m

4 7 18 5 5 8 4 4 7 18 5 5 8 0

m,

0 04

‘

his s m all d i ffere ce is quite egligible but i f i t w ere to be at t e d e d to w e re m ark t ha t 1 e c s will o t d i ffer se sibly fro m c s al eady calculate d d w e have 1 T

n

n

o

n

e

,

o

r

u ,

,

m — ml 1—

e co s u

,

n

,

n

an

m l —

u

_

c

m1

oos

a ,

= 10 1 2 0 33 12 hus fi ally = 1 0 1 2 0 33 1 2 we H avi g fou d the ec e tric a m aly subs t i t u t e t his i equatio ( iii ) t o fi d v For this pur pose it is co ve ie t t o write equa t io ( iii ) i t he for m =t ta é t v § w here S 4

an

d t

n

u

n

n

n

n

an

)

n o

n

n

in

'

.

c

n

n

’

°

n

n

°

an

u

.

n

n

u

,

’

“

K E PL E R S ’

’

[

N E W T O N S LA W S

AN D

OH

V II

.

T ables are u e ful as e abli g us at l t h ugh B s hi g o ce t o b t ai a go d a ppro xi ma t io to t h require d value t hey are t i dispe sable A y f the graphic l m e t ho ds woul d rea dily deter mi e t o withi t hree four d egree f t he true value We m y t he b t ai a value as accura t e as tha t f t he tabl es by t he hel p f f ur pl ce l gari t h m s I f fo e a m ple w e have fou d = 1 0 5 by a graphical process the e x t s t e p m y be co ducted as follows A

n

u

O

n

e

a

n

o

-

n

o

u

a

.

n

.

n

n

s

n

n

n

n o

er s

n

o

n

O

n

c

au

o

’

or

o

s

n

o

o

a

o

.

r

,

x

,

°

n

,

a

n

L o g s in

99 8 4 9 5 2 97 5

it,

log e cosec 1 log e i s n u

n

99 8 3 1 n

,

e sin u

”

19 1 600 ° 53 13 3 105 00

,

u ,

m —

m

o

51

46 7

m

47

18 9

mo

— 4

log ( m log ( 1 log A

m, )

n

cos

e

u

,

)

00 96 6

u ,

n

Au

278

=

o

u ,

10 5 0

u

10 14

,

roble ms which arise i the m aj ori ty f cases are t hose i w hich the ecce tricity is very s m all ; fo e xa mple i the m t io o f t he ear t h about the su the ecce tricity is m ore tha For such ca ses i t is best t o obtai appro i mate e pressi f the su s true a o m aly i te m s f m i the form o f a series which eed n o t f m st pur poses be carried beyo d Writi g si I i s t ead o f e we have fro m 5 2 ( iii ) Th e p

n

o

r

n

n

’

n

n ()

n

ta n

whe ce n

if

an

d

a c

f“

(1

= u + 2 (t a n

e press x

or

n

1 ( n w/

ta n

a

kt)

7

a ieria logari t h m

f N p

n

,

s,

)

6

e

4

“

o

f

w

1 e/<

ta n

ta n

s

a n(

f“

m e

ta n

bo t h sides

) si n u

3d

7

+

t an g y

2

5 ¢ si n

2u +

or m ula i t er m s O f t h ecce trici ty e we have

t he f

ya

by sub titutio s

on

—t u /z

n

tan

d

o

(1

ta n

ta n

u

by t aki g logari t h m s TO

an

%

1 ( %

ta n

v

m e n(

v

x

n

e

s

n

x

o

n

n

e

“

an

r

o

n o

o

be the base

e

n

v

or

n

n

n

n

n

n

n

1 (

VI

e

n

,

w

n 3 1 — 2 u + 1 7 e 8in

3u

( v)

.

K E PL E R S ’

N E WT O N S L A W S

A ND

be the i crease

If d m

t he

in

n

[

’

m

ea a o m aly n

V II

.

the t i m e

in

n

OH

dt

t he n w he n ce

equa t io

dm

r

dv

a n

(1

dv

(1

he ce

e

2

% )

cos

e

v

)

2

n

m an

b

be w ri t te t hus

n

dm w

2

d

(1

(1

2e

3e

co s v

4e

fl

2

cos v

3

cos

3

v

) dv

,

by i tegratio n n

si + Qe w here po w ers f e above the third are eglected E T f a l his series be obtai ed as ollo w s G e er n s i on c [ p We h a ve fro m ( viii ) m=v

2e

.

’

n v

3 — i s n 2v ge s in 3v r

o

*

n

x

n

a

f

dv If

we

m

ake

v .

si n

d

a

1 + si n

f 1 + si n ¢ c o s v

>

whe r e

e

—

4”

Sl n

cos kv

22

{1

he ce

t a n lflt

1

4)

tan ta n

d

¢> c os v ¢

.

it is easy to verify that

=

s i n c>

an

n

an

.

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d souther bou di g circles the To r r i d Zo e d i t s or t her are ter m ed t he Tropics f C a cer d C apricorn respec t ively orth d south are called t he T h e parallels f lati t u d e 66 33 Th o e i clu de d b d A t c ti c C ircles respectively A c ti c twee t he A rctic C ircle d t he Tropic Of C a cer is k ow as the N or t h T e mperate Z o e w hile t ha t bou d ed by t he T pi o f d t he A n tarctic C ircle is t he S outh T e m pera t e Z o e C a pric r d S outh P oles bou d e d by L astly t h e regio s rou d t he N orth the A rctic d A t arctic C ircles res pectively are k o w a s t he N orth a d S outh F i g i d Z o es 23 27 d w e have A t t h ti m e o f the su m m er olstice 8 A t t he f oi t the rctic C ircle n c t an 8 = 1 d er U p yp these circu m s t a ces the hour a gle f t he a t risi g or setti g is a T hat is t o say t he diur al course o f the n is the circle parallel to the equator t ouchin g t he hori o at t he orth m f f d oi t so tha t at i igh t hal its isc w oul be visible w e d o d ( p are n t here taki g t he e ff ec t o f re frac t io i to accou t ) Withi the frigid o e t he s will rem ai above t he hori o without setti g f r a co ti n ually i creasi g u mber o f d ays as the observer approaches t he pole To observer at t he pole it sel f the s w oul d appear to m ove rou d the hori o at t he equi o a ft er which i t will describe a spiral rou d d rou d the sky gradually i creasi g its height above the hori o u t il at t he solstice i t s d iur al track will be very early a rcle parallel t o the hori o a t altitu de o f 2 3 A fter the solstice it will retur i a si m ilar spiral curve t o w ard t he hori o w hich it reaches a t the autu m al equi ox I t h e w i t er hal f o f the year t he s will be co ti uously belo w the hori o m i m he o e a the south t e erate south rigi Th d f d p p o es w ill be si milar t o those i the correspo d i g orthern o es but they w ill occur a t o pposite e pochs o f the year T hus the spri g o f the souther he m i phere coi ci des i poi t f ti m e with autu m i the orther he m i phere t he su m m er o f the d i c e ve s a N orth wi t h t he W i ter f the S outh I the torrid o e t he co ditio s are as follows O t h e equa t or si ce 0 whatever m y be 0 we ha e fr m ( i ) cos h be t he value o f 8 H e nce h = $ or t he le gth o f the d y is 12 hours all t he year rou d e ith dista ce f T h e m eri d ia t he will ho w ever vary fro m d y to day A t the vern al n

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tu des by the fac t or fo the or t her he m i phere No

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tr o d u c t o r y W e have alrea dy lear ed tha t i co seque ce f at m os pheric re fractio there is ge erally a differe ce bet w ee t he true place o f a celestial bo dy d t he place w hi h t ha t body see m s to occu py W e have here to co sid er a other d e a ge m e t f t he place o f a celestial body w hich is d e t o the fact t hat the velocity o f light though doub t e tre m ely grea t is still t i co m parably grea t er tha t he vel city w i t h which the observer is him sel f m ovi g A y a ppare t cha ge i n the place o f a celes t ial body arisi g fro m t his cause is k ow as a be ti T h true ordi a t es o f a celestial bo dy ca o t t here fore be ascer t ai ed u t il certai correc t io s f aberra t io have bee applie d t o the appare t coordi a t es as i dica t e d by direc t b e t i l Th e a t ure f these correctio s is w t be i ves t iga t e d r d th w h di c 1 T h t t hi m u t b t h p rc i d b y B m r wh f l i gh t i 1 67 5 T hi l t r h wr t t H uyg g du l pr p g ti pp r i mp le d 0 H yg T m p T h ugh (O g i p ri dic h f t h P l S t r r ll y d t h pl u c d i 168 0 b y t b rr ti w P ic rd t h cr d it f di c ri g t h g r l ph m f b rr ti i d t B r dl y wh l g t h c r ct xpl ti f it 79

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,

s

a re

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o

.

= 0,

which h w th at tar I g eral t h re pecti vel y

c o s c»

f the

d 80 ,

.

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n

e R A

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l i quity f t h ec l i ptic If t h i ab rrati i tati ary th e d i ffer ti al c effici e t f ( 3) w h ave ob

i s the

w

.

a)

n

n

e su n

o

ta n

where

.

n

a

x

n

80 t a n

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(1

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tan

a)

1

.

2 2

2

s

.

on

256

TH E ABE

Ex 2 .

tude

o

If

.

f t he

ab rrati e

8 be t h e

a,

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d

ecl i ati

ha s

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(0

(

ap x

i f A be the

HT

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of

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e,

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co s a

.

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d

an

e

o

s

ab errati

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n

OF

l i uity f t h cl i ptic h w th at whe i t g rea te t val u

b q

o

cl i ati

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on

n

8 1 , E x 1, t h e

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i i m um it m u t b t h m u t th eref re c ta i K fr m t h t r i a gle S K P P r ve th a t w he t h ab errati 3 attai i t gre te t E i de c l i ati um rical val e f t h year t h arc tar t h cele ti al phere j i i g t h at ri ght a gl d t t h p le f t h e eq u a t r t th 4 P r ve th at f a g i ve p iti f t h E t h ab e r ati o i ri ght a ce i f a tar t h quat r wi ll b lea t whe = t a 6) t f th tar 6) t h u l g itude a d t h b e i g t h r i ght a c e i b l i quity f t h ec l i ptic 5 P ro ve th at all tar wh ab rrati i R A i a maxi mum at E ame ti m e th at t h ab errati i d cl i ati va i he l i eith r th c e f t h c d rder wh e ci rcular ecti ecl ipti e parallel t o t h l titi al c l u re th d equat r [ Math T r i p ] i d ec l i ati o i er w hav A t h e ab errati = ta 8 = t t i 8 ) ( wh c e =t t 8 8 i ta ) /( ta t t t 8 =t 8 8 2t 8 si +t /(t a i R A i a m ax i m u m w h ave (E 1) B t a s t h e ab errati m

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l titial c l ure which ati fy

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p

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OF

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[

a gre t ci rcle three p i t R R R b t ff at d i ta ce p t h ci r c le re p ctivel y fr m r i gi d if ; y y R re p cti vel y fr m t h ce t r f R R c i b t h d i recti ph ere th e If

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p2 , p3

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apply thi

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t o the p — =x Z 3 p p

s

(P3 ( P3 — (p3 p1) + z3

) + 9 2 si n z2 S i n + 3 ) p

-

n

s

si n

s

9;

,

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we m

—

si n

ake

—

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= P2) 0 ,

) + 303 s i n (PI PI ) + 9 3 s i n (PI PI

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9;

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(pl

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= p2 ) O

-

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.

tr y o f a n n u a l a b e r r a ti o n W e w i vestiga t e t he sha pe f t he sm all closed curve which t he star a ppears t o d escribe t he celes t ial s ph ere i co seque ce ual aberra t io f a L e t S T be t h per pe d icular fro m S the true place f the where A is t he a pe x ( Fig s t ar t o t he eclipt ic d 86

.

T h e ge o m e

.

o

n

n o

on

n

n n

o

n

n

n

.

e

n

o

,

an

.

,

o +

F IG

produce

.

90

°

72

.

to T so t ha t S T = L t S be the poi t to w hich t he s t ar is dis place d b y aberra tio the n si ce SS i s m all w e m y regard the locus o f S as a f I d o S la e curve the recta gular coor i ates as f x b e p y i dica t e d i the figure we have 8 1) e

n

T

’

’

’

n

’

n

,

s

’

a

’

n

n

.

n

n

n

,

,

x

i

x s n 2

x

S A si n A S T = 2

y

cosec B 2

1c x

cos ( CD

2 .

A) ,

8 5 87]

TH E

-

A BE

RRA TI O N

OF

L IG

HT

We t hus obtai the followi g resul t s with regard t o the e ff ect f a ual aberra t io the appare t p sitio f a s t ar I co seque ce f a ual aberra t i t he a ppare t place f each star describes elli pse k ow as the elli pse f aberra t io i the course o f a y d t he ce t re o f t he elli pse is t he true place f t he s t ar Th a is m i or f the elli pse is per pe d icular to the ecli ptic T h se m i a is m aj or f t he elli pse is t he co t a t f aberratio l l stars d is t h ere fore the sa m e fo For a star o the ecli pt ic the elli pse beco m es a straigh t li e For a s t ar a t the pole f t he ecli ptic the elli pse beco m es a circle d i ge eral t he se m i a is m i or o f the elli pse is the pro d uc t f the si e f t he s t ar s la t i t u d e d the co sta t o f aberra t io E 1 A u m i g th a t t h u m ti i u if rm h w th at at f ur c cuti v ep ch at i terval f th ree m th t h appar t pl ce f t h tar w i ll ccupy ucce i v l y t h f ur ex tr m itie f a pa i r f c j ugate d ia m eter f th ll i p e f ab rrati 2 E L t A b t h l g itud e f a t r d 8 i t latitude hwg wi ll b t di plac t h tar b y a m et ricall y th at t h e ffect f ab er ati d i ta c w hich i t h qu are r t f n

n

n

nn

on

n

n

n

n

o

x

.

n

n

n

o

o

n

n

,

o

.

n

o

x

-

o

n

n

e

n

on

,

earan

e

o

nn

an

o

-

.

o

ns

r a

an

n

o

n

.

n

n

.

o

n

an

,

n

x

-

n

’

o

o

n

x

.

an

n

ss

.

e

o n se

s

o

x

s

o

e

s

n

on

oo

.

S

.

h w th at

the

o

s

,

o

e

s

o

en

a

o

o

o

e

on

s

an

e

,

s

o

e

s

e

o

eo

s

o

5x {1 Ex 3

s,

on

?

1

o

n

.

.

s a

r

s

e

s

e

o

on

e

o

s

e

on

o

e

e

s

o

on

e

e

o

e

e

.

.

ss

e

n s

s

s o

n

s

o

s

e

’

n

n

n

3 co s 2 ( G)

elli p

of

se

aberrati

rth g al pr jecti ta ge t pla e t uchi g t h

i s t he

on

o

o

on

o

on

a ci rcl i t h pla f t h ec l iptic t h cel ti al ph e i t h t rue place f t h tar ual ab errati up t h app r t place E 4 S h w th at t h eff ct f f t h fi xed tar w u ld b pr d uced i f each tar actuall y rev lved i a mall ci rcular rb it parallel t t h pla e f t h ec li ptic d i f t h earth were at re t E ff e c t o f th e e l l i p t i c m o t i o n o f th e e a r t h o n a b e r r a t i o n 87 We have o w t c si d er the i flue ce f the ecce n trici t y f t he ear t h s orbi t t he a ual aberratio L t G) be as usual t he su s geoce t ric lo gi t u d e the 1 8 0 G) is t he ear t h s helioce tric lo g itu d e the lo gitu d e f perihelio Th e ear t h s d 9 the t r ue a o m aly s t ha t G) ra dius vec t or is ha e t he s i g i fic t i o alre a dy d if v 8 give to the m w e m ust have o

f

er

s

es

e

n

e

o

ne

on

e

e

n

o

e

e

s

n

n

o

n

n

e

.

.

x

o

.

e

o

.

s

e

s

o

e

o

e

o

o

o

a

e

e

en

s

n

o

s

n

e

on

on

an n

o

s

s

e

an

.

*

.

.

n

o

’

on

on

n

n

n n

n

n

e

’

o

o

.

n

n

°

n

,

’

n

a n

n

n

r

,

an

,

a

o

n

:

n

’

o

0,

,

010

v

n

a

n s

n

v

cos 8 cos a

v co s

0

g

“ t

,,

cos

i

80 s i n

a0

v si n

80

_

dt 1 7— 2

2 60

TH E

ABE

RRATI O N

OF

LIG

HT

first o f t hese equatio s is obtai e d by i d e t i fyi g tw o e fl T ressio s the velo i t y f t he ear t h arallel to the li e f p p T h t hir d equatio is ob t ai ed from ide t i fyi g e pressi s f d t he veloci t y f the ear t h parallel t o t he ear t h s polar a is the seco d equa t io is obtain ed i like m a er fro m the a xis d m er e icular to t hose alrea y e t io ed d p p T o m ake use o f equa t i ( i ) w e m us t b t ai fro m the elli pt ic d of Ke pler s m o t io the values f d /d t seco d law ho w s t ha t d9/d t 1 / ( § d hen ce fro m the = a 1 i olar equa t io t he elli se c s w e f e + ( p p ob t ai Th e

n

n

n

n

n

n

e

n

o

c

or

x

n

n

n

x

o

n

n

n

o

oc

o

n

’

“

r

s

n

n

o

an

r

an

.

ons

n

,

n n

n

n

or

on

x

’

n

.

r

v z

,

an

o

r

.

n

r

d9/dt

C (l

e cos

where C is a co sta t ; by substi t u t i g t his i the logari t h m ic di ffere tial o f t he polar equatio f t he elli pse w e fi d n

n

n

n

n

d r /dt

a di g ( i )

E xp

n

n

an

v co s

8o c o s

a,

v co s

8o si n

a0

v si n

80

d m

n

,

Ce si n 9

.

n

on

Oo s w

C si n

_

o

aki g these subs t ituti s C {—

= C

n

w

9 c o s G) + si n G)

e si n

1 e + (

— e s i n 9 s i n G) { — e s i n 9 s i n G) {

whe ce re m e m beri g that 18 0 G 9 w e obtai 8 cos 0 ( s i n G) e si ) i C o s w ( — cos CD + e cos v cos 8 n

°

n

,

v co s

a ,

0

a,

0

s n a,

80

v si n

n a

n

ns

x,

a

’

—a

u se c Ic e

— 8 8= ’

1c

(

a

C sin

ubstitu t in g i equatio which is cal le d t he C/n S

8 ( — si n

sec 8 (

c o s w s in a si n

w

(

—

( i)

co s

G) +

d

ii ( )

an

co n s ta n a si n

t

f

a

o

e co s w an

be r r a ti o n

G) — c o s

a co s

cos cos

s i n a si n c :

0:

8 c o s G) — si n

w co s

a

®

(+

si n w c o s w c o s

8 + si n

d m

aki g n

,

co s w

cos

co

)

)

,

8 c o s G)

cos xe

n

,

c

—

cos

w c o s a si n f

a si n

8 s i n (9 )

8

cos si i A e is o ly about it is plai t hat t he ecce trici t y o f t he ear t h s orbi t h a s but a very s mall e ffec t o the aberratio Th e f t ha t e ff ect is ho w ever wor t hy f otice peculiar character T h ter m s i a d 8 t co t ai G) 8 which co tai e d C o seque tly t hese t er m s d o t cha ge duri g the course o f t he c o s co

s

n

n

a

n a s n

n

’

n

n

o

e

n

n

n

’

o

.

n

.

'

a

an

n

o n

n

o n o

n

n

n

n

.

TH E A B E

RRAT I O N

OF

L IG

HT

[

OH

xI

.

l ossible at t he e i t h a i tt le or t h t he o t her a li tt le d p south f t he e i t h T h s t ars are t o be chose so that their righ t as ce sio s d i ff er by abou t 1 2 hours Th fi s t observatio s e i t h d is t ce o f b th s t ars are to be m a de a d y w he S of has i t u ppe r cul mi a t io a t 6 A M d S will o t he sa m e d y have i t u pper cul m i a t io a t GP M These are to be co mbi e d with observa t io m ad e i m o t hs la t er whe S cu l mi ates a t T h ese co d i t io s d S at 6 A M har d ly be e a tly 6P M reali e d but they i di a t e t he m os t per fec t sche m e f ac ura t e resul t whe o ly tw o s t ars are use d Th e reaso s f t hese require m e ts will prese t ly be m ad e clear L t values f t he righ t asce sio d 8 be t he m ea decli atio f S fo the begi i g f the year take fro m so m e s t a d ard catalogue E ve t h mos t e celle t d e t er m i a t io s o f s t ar places m us t be presu m e d t o be i som e d egree erro eous f t he coor d i ates are very s m all N d oubt t he error d fo m s t pur poses they m y be quite overlooked B t such m i ute errors as are u avoi dable i t he d ecli a t io s ad o pte d f t he stars w oul d be quite large e ough t o vi t ia t e a d eterm i a tio f t he coe fficie t f aberra t io which depe d e d the decli a t io I t he prese t m ethod t h e observa t io s are so co m bi e d tha t t he d ecli a t io s d isappear fro m the resul t d co seque tly their errors are void f e ff ect We sh ll assu m e fo t he m o m e t that a value o f the co s ta t o f aberratio is a ppro i m ately k ow We m y f e a mple t ake t he co sta t t o be where is som e very s m all frac t io f a seco d f Th deter m i atio is the the obj ect f the i ves t igatio B y this d evice w e secure the co ve ie ce t hat t he qua n t ity sought is very m all i c mpariso with the t otal a m ou t f aberra t io d co se que tly i com puti g t he is to be m ultiplie d we are per m i t ted to use e ffi i e n t s by w hich m f ro i ate m etho d s that would be vali d i f t hese coe ficie ts t pp w ere to be m ul t i plied by y qua t i t y o t her tha a very s mall n

z

n

.

d

e

l

n

n

n

,

n

n

n

ca n

x

or an

c

n

or

o

n

r

n n

n

.

n

n

o

n

e

x

o

s

o

n

n

n

n

n

n

n

n

or

n

n

n

o

n

n

n

an

n

r

x

n

n

o

n

K]

n

n

n

n

O

n

,

or

x

,

n

o

n

x,

o

n

n

o

n

an

n

n

n

n

co

n

n

xl

x

n

no

an

on e

,

.

c

a

a

.

x,

s

n

n

n

,

n

e

.

n

n

.

a

n

.

n

n

n

n

n

on

n

O

o

n

n

n

n

r

u

.

.

an

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a

n

an

n

n

O

c

.

1

n

a

I

.

I

I

n

n

o

n

.

n

,

a

c

n

n

n

2

n

.

n

,

.

s x

ns

2

an

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n

n

r

on

n

s

an

e

o

n

z

n

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an

n

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n

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n

.

fir t o peratio is t o de duce the appare t places f S d S f the d ays f observatio W e m ust by the k ow processe co mpu t e t he preces io W e m us t fur t her calcula t e d utatio the aberratio usin g the appro i m a t e value f the e ffi i e t T h e correctio thus ob t ai e d fo the d ecli a t io o f S Th e

2

n

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or

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.

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or

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n

.

I

n

r

n

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1

§

8 8]

T H E AB E R

RA TI O N

or

LI G

HT

t he firs t da y o f observati o n w e de n o t e by p I t is a com ple t e corr ec t io n e x ce pt i n so fa r as we have use d a n i n correc t value o f t he co n s t a n t o f aberra tion W e m ust t here fore i n crease p by A x w here A is the coe fficie n t o f vp as give n i n equa t io n 84 T hus we see that the a ppare n t d ecli n atio n o f S o n the first d a y o f observa t io n i s 8 + We a ssu m e however as above p + A 1c e xplai n e d t hat there m a y be a n u n k n o w n error i n 8 L e t 2 be t he observed z e n ith d is t a n ce w hich we shall su ppo s e cleare d fro m re fractio n ( Cha p T he n si n ce t he latitu d e gb is t he s u m o f t he z e n it h di st an ce ( i n this case su pposed t o be south ) a n d the d ecli n a t io n we have on

,

.

l

1

.

l ,

“

I

.

.

I

1

1

1

1.

,

,

1

,

1

.

,

.

,

=

¢

81 + pl + A 1 K 1

+

Zl

t he sam e day abou t 1 2 hours la t er we O bserve the s eco n d star a n d as i n t hat ti m e th e latitude will n o t have chan ged On

,

,

,

a ppreciably we have also ,

,

where by t he chan ges i t he su ffi ces we i dica t e that this for m ula rela t e t t he seco d star S i x m o t hs la t er t he bserva t io s are t o be re pea t ed t he sa m e star a d we m ust the su ppose the la t i t u d e h cha ge d t o p w hi h ge erally diff ers fro m p Th a cou t f cer t ai m i u t e perio dic al t era t io s e ith d is t a ces are di ff ere t at t he seco d e poch o f observatio d so are also p p A A but 8 n d 8 bei g the m ea values o f t he d ecli ati s a t the begi i g f t he year are the sam e a t bo t h e pochs Usin g acce ted let t ers to disti gu i sh the qua tities relati g to t he seco d epoch we thus have n

s

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n

.

on

as

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8,

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.

p2

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lc ,l

A ge

,

p

— 2

p

'

i

d d

+ 192

,

fo r

n

n

— 7 +1 1

n

n

we easily obtai the followi g equatio n

t here fore the aberra t io n is

u m era t or a he ce is fou d

Th e

an

,

z,

f

n

z

n

n

2

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p

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n

n

s,

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O

,

(A xl

I

,

A2

AI

,

A2

1

)K

n

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w here

e o mi a t or are both k ow qua tities n

x,

n

n

n

,

an

d

TH E

AB E

RRA TI O N

OF

HT

L IG

[

OH

.

x1

fp p o allowan ce had bee co m pu t a t i p aberra t io the f rm ula j us t give would have a ff rd e d m ad e f appro i m a t e value f the aberratio d the a ppro i m a t e m y be egar de d as havi g bee t hus value we h ve use d ob t ai e d We are t o o t ice t hat 8 d 8 have bo t h disappeare d If t here fore t h ese qua t itie h d bee a ff ec t ed by s m all errors as f course will ge erally be t he case t hose errors will t have i m par t ed y i a ccuracy to e ce pt i so f as A p St A s 4) d 95 have al o bo t h d e pe d o t he ado pt ed values f 8 d u cer t ai t y as t o the la t i t u e a t either the d i a ppeared y firs t e p ch or the las t will also have very li t tle i flue ce I t is by the obs erve d qua ti t ies t hat errors f observatio are i t ro d uce d i to t h e pression f H w f d e pe ds u po t hese err rs will i flue ce t he value f t he de o m i a t or A — A Th e larger t his d e o m i a t r t h e larger will be t he qua ti t y by which t he errors will be d ivide d d co seque t ly t he s m aller will be the i flue ce f the errors o f bserva t i the resul t Th observa t io s are there fore to be arr ge d so as to m ake t his d e om i a t or as grea t as circu m s t ces w ill per m it To dete m i e t he m s t suitable arra ge m e t w e m y appro x i mate value f A A A A though f course the t rue values m ust be use d i the actual d e t er m i a t io f our prese t A t he stars cul m i ate ear t he e i t h we m y f obj ect su ppose t ha t t heir d ecli a t io s are equal t o t he la t i t u d e d 84) d t hus w e have A si 8 cos p cos 8 i 4 cos ( ) cos A cos 8 si si n 8 cos t 8 ( q A cos 8 si 4; {cos ( cos ( A 2 cos 8 i < i i ( ) 4( 4 1 I like m a er A A; 2 cos 8 i p i } ( ) si } ( w here f the a pe at t he t i m e f the seco d 8 is t h e positio observa t io A s the a pe x is t he eclipt ic cos 8 d cos 8 have as e t re m e d li mi t s W e shall t here fore t ake w ith su fficie t If i n t h e

,

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),

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) s n

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a2 .

a,

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T H E AB E

RRA TI ON

[

O F L IG H T '

GH

xi

.

the la t itude p t he veloci t y f the observe r arisi g from t he earth s rota t io is 4 63 s p m etres p seco d d as t h veloci t y f ligh t is abou t kilo m e t res pe sec o d we t hat t he coe ffi cie t f d iur al aber atio is 4 63 cosec 1 cos gb/30000000 0 cos (p m y al w ay T his coe fficie n t is so s m all tha t d iur al aberratio be eglecte d e ce pt whe grea t refi e m e t is require d carries the bserver toward s t he eas t T h e diur al ro t a t io — h oi t the hori o H e ce d 0 h w here f 9 + p is t he w est hour a gle f t he star M aki g t hese substitu t io f t he s t ar w he d decli atio i § 8 4 w e fi d t h a t the R A ff ected by d iur al aberratio beco m e cos cos h sec 8 8 cos 4 si h i 8 Whe a star is o the m eridia h 0 n d the e ff ect o f diur al aberratio i d e cli a tio va ishes w hile the t ra si t is d elayed by t he a m ou t cos 4 sec 8 For lower m erid ia tra si t s d the tra sit is accelera t e d by 0 0 2 1 h= cpsec 8 the e ith dist a n e T fi d the e ff ect f d iur al berra t io t he m eri d ia w e di ff ere tiate t h e t f a s t ar which is equa t io i 4 i 8 cos cos gt cos 8 cos h d d 8 the values d substi t ute f dh co s h se 8 p cos gb i h si 8 res pec t ively d obtai d d cos 4 cos 8 i h cot E 1 S h w th at t b erver i latitude p a tar f decl i ati 8 wi ll wi g t d iur al ab errati appear t m ve i ellip e wh e em i axe m 1 dm a i 8 wh re m i t h rati f t h ci r cu mfere c e f t h earth t th d i ta ce d c r i b d b y l i ght i a d y d t h a gle i ci rcular mea ure [C ll E xam ] E 2 S h w th at t h effect f d iu r al ab rra ti th b er ed e ith d i ta c e f a tar m y b all wed f b y u b t ra cti g t ec d f m t h ti me f b ervati w here t i t h ti m e i ec d th at l i ght w uld tak t t ravel a d i t a ce equ al t t h ear th rad iu [ Math T ri p I ] P l a n e t a ry a b e r r a ti o n 90 Up to the prese n t w e have assu m e d t ha t the s t ar w hos e aberratio w as u d er co sidera t io w as itsel f at res t B t i f At

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to t he be g

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RR AT I O N O F L IG H T tar b ec m e 8 + \i w ith ut y ch a ge i t h p iti f t h c rd i ate th at p d d b ei g fu cti 2 74

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sin

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d

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as

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val u

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of

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requi red r l t

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h w th at i f a um b er f t r l i a ci rcle f which t h ar cual radiu i very mall t h eff ct f ab errati th e e tar i t c vey th em adj ac t ci rcle (B ii w ) t t h ci r cu m fere c e f ce f p fr m equati E 3 T hi f ll w fr m t h ab d B b tw tar which appear t b e co veyed b y a be r L t A E 6 rati t A d B t ward ap x C S h w that t h ab rrati o cha g — t A l a t A i t th g § i p w here i t h arc A B a d p i t h AB perp d icu lar fr m C tar at B d A b at d i ta ce L t th tw b re pecti vel y fr m 0 i th Th ph er ical t r i a gle w h ave b C= i b t i C ot A d i ffere ti ati g a d maki g *

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si n

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n

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tar pr d uct i s very

A s i n b,

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ta n 4 5 0 si n p

.

all that whe mul tipl i d b y t h m all d A A b ec me i appreci ab le E Fr m t h tar defi ed b y ( = 5 9 t h d i t a ce 7 d p iti a gle 2 0 7 14 f adj ace t tar were mea ured t b appl i e d t 1880 S h w tha t t h c rr cti th d i ta c e 6t h J d p iti a gle t reduc them t t h date 18 7 9 0 d 0 6 6 re pecti vel y If th e

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TH E AB E

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=

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a ci rcle with erv r t h earth u rface d t i t f i accu rat el y g i ve b y t h y tar

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E XERC SE S O N C H A P TER X I Ex 1

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L IG

L i ta ce L g 1 t term = L g ab rrati i 9 0 36 p iti e y ar i prec

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on

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OF

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have f

m N A 18 80 , p 30 3, w e .

RR AT I O N

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wher 0 i a equat r d i fferi d h t h rati

sin

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00

s i n (T E c o s

ecl i ptic

ehi

0 0 E + n 2 s i n 2 o E )’f '

a

i t o th b y t h c m plem e t f t h h ur a gle R A fr m t h g i f t h vel city f t h earth c e t r t t h vel city f l i ght [ Math T r ip ] 0 0 f le gth b e d raw fro m t h vertex If i a ph r ical tr i a gle egme t B 0 = l d A 0 = m the d iv i di g t h b a e i t t w po

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tar b at C d i f B b e t h apex f t h r tati al m tio d A th at rb ital d i f x b t h r ul ta t ab rrati th u i x = p i ( — x ) o f th where p i t h re ul ta t vel city f t h e b erv r d p t h vel city f l ight d w h ave c cl c ec m p c ec ( l If t h e

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tan

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the

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.

h w th at t h l cu f all tar wh e e ith d i ta ce at a g i v d a g iv i ta t u al t red b y ab rrati i elli ptic c e pl c e i h ri f w h e ci r cular ecti tal d t h ther i perpe d icular t t h ecl i ptic Ex 2 .

a

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18 — 2

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iti f r fr cti d i ta c e f

po s

RRATIO N

a gle ub te ded b y t h e ith d t h apex at t h tar fr m w hich t h d i red r u l t i ea i l y b ta i ed Pr ve th at at every plac e th ere i alway a t a g i ve i ta t a tar f which t h ab rrati i e ti rely c u t r ct d b y t h th at at m id i ght th h rte t d y t h e ith S h w al thi p iti i gi ve by equati f t h f rm the

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z n

o

1,

2

c rrecti f r fracti b a u m ed pr p rti al t t h ta g t f e ith d i ta ce d t h earth rb it b ass um d t b ci rcular th [ Math T rip I 1 900 ] E 4 If b y th equat r t h e c rd i ate 8 f y m all ch a ge i each po i t t h cele tial ph re b c m 8 + d h w th at w m u t h ave

i f t he

o

on

s

z n

e

or

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ss

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e

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.

.

n

on

e

s

n

d

d

z

h re A thi t ra e

s

,

B, C

n s fo

a re

rmati

c

on

o

G+ A s i n

(u

e

>=

w

e

s

s

n

=A

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s

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s

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+ B ) t a n 8,

ta t i depe de t f t h c rd i ate leave t h d i ta c e b etwe ev ry pai r

on s

o

.

oo

r,

en

e

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n

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v ri fy that tar u al tered

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n

.

TH E

G EO C E N T

R IC

PA

RA L L AX

MO ON

TH E

OF

[O IL

XI I

geoce tric parall a o f the s is t he a gle OS O w here S is t he ce tre f the d C C p is t he earth s ra d ius re prese t t he a gles S O Z d S OZ respec t ively The the e ff ect f paralla m y be sai d to throw t he a ppare t place f t he obj ect away fro m the directio 0 0 by the a gle C — C which w e hall re prese t by t he sy m bol 4 O f course i f the earth were regar d e d as s phere the C a d C woul d be the appare t d real e i t h d the i flue ce o f paralla would m erely depress d ista ce t he a ppa e t place o f the bj ec t fur t her fr m t he e i t h A t he earth is t s pherical t he e ff e c t f paralla is t o d epress the b dy e i t h but fro m t he poi t i which t he o t e ac t l y fro m the earth s radius whe co t i ued will m eet t he celestial sphere Th e bet w ee this poi t d t h t rue e ith is o f course t he qua t ity already co si d ere d i 15 Fro m t he tria gle OS O we have Th e

x

n

an

n

’

n

77

a

n

,

an

s

n

’

,

.

n

n

n

s

.

x

o

z

n

z

n

z

o

,

x

an

x

O

n o

n

s

n

n

r

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n

.

a

x

o

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an

,

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n

n

n

’

su n

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u n

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o

n

n

n

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n

ar

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n

n

.

n

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an

n

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n

.

'

n

sin W

We an

d

n o

w

i troduce the a gle n

n

he ce fro m ( i ) n

C/r ’

p si n

; 7r

d

¢

efi e d by t h e equatio n

n

p/

s i n 7 4,

H

r

,

si n

C ’

s i n 77 4» s i n

.

hus we see that m, is t he greates t value o f w ; d this w ill be attai e d w he C is 90 which i f re fractio w ere t co sidere d w oul d m ea that the ce t re f t he su w as o the hori o We accord i gly t erm 1, the ho i o t l p l la x A s the hori o tal par llax d e pe ds show ii s i d p ( ) as p is o t the sa m e f all latitudes o wi g to the s pheroi dal for m o f t he ea t h it follo ws t ha t the hori o tal paralla m ust vary w ith t he latitude o f the observer Its m ax i m u m value is a t t ai ed whe the observer is the equa t or d as (p is the ero w e e press by what is k o w as the e q to i l ho i o ta l l f f d a l a x so that i is the equa t orial ra ius o the earth w e have p p T

an

’

°

n

n

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n

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n

o

r z

71

a

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qr ,

n

n

an

u a

r z n

r a

0

,

si n 7 0

p/ o

r

.

the u n be t its m ea dis t a ce so tha t equals a t h se m i is a is m aj or o f the su s appare t orbi t the t he qua t i t y defi ed t o be the me a e q to i a l ho i o ta l p a l l a x of the s d is gi v e by t he equatio If

s

a

’

x

n

n

u a

r

W e shall a l w ays take

r z

W .,

pO/a

.

n

n

,

n

s i n vr a

e

r

n

n

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ra

7 rd

u n

,

§

92 ]

R

TH E

G EO C E NT I C P A

RA L L A X

MO O N

TH E

OF

sy mb ls already g ive apply to t he geoce t ric paralla f B y t he addi t i t he o f a d sh t o we d e ote the c rre s po di g qua t ities f the m oo thus i l h t he i l m a e o en t c o t e o o i the a gle w hich c e g g p f t he ce tre o f t he ear t h d the positio f t he O b erver sub t e d a t the ce t re f t he m oo is t he a gle whose si e is t he ra t io f t he d ista ces o f the f the earth obs i d t he m oo s ce t re fr m the ce t re l i T his is t he h o i o t l p ll h e m o o t a t t de t f t he equator is t he value o f W t w he the bser ver is t his is the e q to i l h o i o t l pa ll x of the mo o w he t he m o is a t its m ea d is t ce is t he value f T his is t he m e We eq t o i a l h or i o n t l p a ll a x of the m o o shall t ake m T h e m oo is here regar d e d as a s phere d the se m i vertical a gle f the co e w h ich this sphere subte ds at t he earth s ce t re i e the a ppare t se m i d ia m eter o f t h m oo varies fro m 1 6 4 7 to 14 d has a m ea value f 1 5 From the for m u l a ( ii ) w e ob t ai p cosec T h radius f the ear t h is a k ow qua t i t y is also d if k o w t he i t his equa t io t he right ha d side is k ow d T hus w e ob t ai the i mporta t result that t here fore is k o w t he dis t a ce f a celestial body be de t erm i ed whe its hori o tal paralla is k o w I t is i fact o ly by deter m i i g t he parallax f a celestial bo dy by observa t io that w e c ce t ai its dista ce d a s the de t er m i a t io f these d is t a ces is f the ut m os t i m por t a ce i A stro o m y it is bvious t hat t he subj ect f paral l a m eri t s care ful atte tio too T h e g e o e t i c paralla O f a st r pro perly so calle d is f the case o f eve the eares t star m i u t e to be se sible I d ( C e t auri ) t he hori o tal paralla w ould be o ly could be d etected by our m easure m e ts w hich was o paralla o t m ore tha a t housa d tim es grea t er tha t his qua t ity I t is there fore i mpossible to determ i e t he dis t a ce o f a s t ar by i t s geoce t ric paralla For such i vestiga t io we have t resort to al p a l la x d the co sideratio f this is de ferred t o m rese t roble is t hat f geoce tric paralla C h p XV O p p Th e

su n

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RIC

G EO C E NT

TH E

R AL L A X

PA

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OF

[O H

.

M I

es pecially i its a pplica t io t o t h m o the m ea equato ial I C ha pt ers X I I I d XI V hori o tal paralla o f w hich is 5 7 d other we shall co sid er the geoce n t ric paralla f t he s bodies i the solar sys t e m S h w th at 1 E d

an

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mo o n i n

f

r

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(1)

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tai

A 8 + OO S

re e t ca e s

n

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s

A8 =

'

1r 0

if

n

be t he

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co s

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1r 0

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i

= 0,

Aa

1r 0 s n z,

s 7; s i n z + c o s

( co

si n

an

4) s i n

h

cos ( ) s n

[ Co ll

e ith d i ta ce

h A cp

s

o

e s se

s

a s se e n

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n

z n

o

s

s

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7r 0

en

s

) co s

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.

In th e p

e

-

h o w that i f t h h ri tal parallax b e a qu a tity w h e eglected t h appar t dai ly path f a cele ti al b d y y b earth rface ( upp ed ph erical ) i a m all circle f rad iu i (1 8 de c i b ed ab ut a p i t depr d 1 i 8 b el w ’

8+

t h e po

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s i n 7r

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h w th at parallax i crea t h appare t emi d iamet r f t h e i (C where C i t h appare t e ith d i ta c i C t h rati eart h i a umed t b ph rical

Ex 2

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t o 2 m i s t he q

ua tity requi r n

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.

l c ul a ti o o f a n e c l i p e T o illustra t e the for m ulae w e hall co m pute the total ecli pse f the m oo w hich oc urre d Feb 8 t h 1 906 follo w i g are the d ata ( see N a ti c a l A l m a Th 1 906 c 119

.

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4 83) p .

Th e

ep ch or G ree wich m ea ti m e f co j u c t io f m oo d ce tre f sha d o w i R A is o

n

o

n

n

n

n

o

n

n

.

.

an

o

n

11

19

49

In

59

3

1 1 8 — 11 9]

MOON

E CL I P S ES O F T H E

ight asce sio f m oo t epoch M oo s decli a t io at e poch 8 D ecli atio f shadow a t f ce tre e poch 8 H ourly m otio O f m oo i R A d H ourly m otio o f sha d o w i R A H ourly m otio f m oo i d ecl = 8 H ourly m o t io o f sha d o w i d ecl M oo s equatorial hori o tal paralla x S u s equa t orial hori o tal paralla M oo s a gular se m i dia m eter S u s a gular se m i d ia m e t er e R

n

n

o

n

=

a

h

’

n

n

n

n

14

n

o

n

n

n

n

.

n

n

o

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n

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n

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.

D

28

2

29

7

42

48 1

9

r,

r

-

in

18 3000

2

34

x

t hese values 34 8 ) we ob t ai n

(p

24

.

n

S ubstituti n g

given

16

55

.

-

n

’

n

z

z

14

.

.

n

’

n

48

.

n

n

n

n

°

22

“

o

’

n

28

9

a

m

15

47

16

13

t he e xpression s fo r A

3 5 4 0 00 t

’

,

B , C a l r e a dy

36100 0 09 ,

w here D is the d istan ce i n seco n ds o f a r e between the ce n tre o f t he m o on a n d t he ce n t re o f the S ha d o w w here t is the ti m e i n hours si n ce the e poch a n d w here n o t m ore t han t hree Sign ificant figures are re t ain ed S olvi n g this equa t io n fo r t w e have ,

,

.

0 4 91 i

we

If

m ake

cos 9

4 1 8 /D ‘

wfi

t his beco m es

,

049 1

i

2 1 98 t a n 9,

that the ce tres f the m oo the d ista ce D at the G ree w ich m ea ti mes m m h 19 4 7 1 i 13 2 t a 9

an

d

e

n

d

n

o

n

n

n

an

n

.

d

the shado w are at

n

'

°

shor t est dista ce D is 4 18 f 9 w ould o t herwise be i m agi ary d the corres po di g ti m e i the m iddle f the m ecli pse is 1 9 4 7 1 T fi d the first d last co tacts with the pe u mbra we m ake The n

,

an

n

h

n

e

o

.

.

n

an

D

n

n

( p, + po

ro

)

r,

cos 9 0 7 60 d d accordi gly the required t i m es are m m m l 6 54 19 47 i 2 5 3 an

an

.

,

'

,

o

or

n

5 4 99,

ta n 9

1 3 1,

n

h

11

h

an

h

d 2 0 40

m .

23

2

M O ON

E C LI PS E S O F T H E

For the first D

a

h

19 4 7

m

For the first shado w

'

re

1

i

an

d

D

)

ro

1

m

50

last

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ta n 9

11

17

°

an

o e ts

m m

8 35 ,

( p, + po

f

ro

h

d 2 1 37

m

'

3

.

i t er al co tact with

of

n

n

n

)

n

1 62 0,

r,

2 58 cos 9 t n 9 37 5 n d accordi gly the required ti m es are h m m h m 18 5 7 7 a d 20 19 47 l i 49 4 a

,

a

,

ti m es are

qi i i r e d 11

35 10

r,

1 1 9,

'

n

xv r

a

n

( p, + p ®

cos 9 n d accordi gly t h e

.

last co tacts w i t h the sh d o w

d

an

[

CH

,

n

’

°

°

h

n

the poi t the m oo s li mb at which first co t ac t with t he S hado w takes place w e have to fi d t he d ecli atio s h m h m T his is 1 5 3 be f re d the sha d o w at 1 7 5 7 0 o f the m oo t he e poch but the m oo is m ovi g southwar d s i decli a t ion a t the rate o f 7 4 2 pe hour H e ce at t he t i m e f first co tact the decli atio f the m oo m ust have bee grea t er tha a t the e poch d there fore it was 1 5 I t his ti m e the s would m ove orth d the shad w there fore south H e ce t he d ecli atio f the ce n t re o f t he sha d o w at the ti m e f first co tac t m ust have bee 14 H e ce fro m p 35 4 we have cos N M T 0 10 2 d the poi t f firs t co tac t is 9 6 fr m the n orth poi t f the moo towards t h e east be T fi d the t errestrial statio n fro m w hich the ecli pse best see w e deter m i e the la t i t u d e d lo gitu d e o f the place the earth w hich w ill lie direc t ly be t wee the ce t res f the earth d m oo at the m i ddle o f the ecli pse h m Th m i dd le f the ecli pse is at G M T 1 9 4 7 1 d there fore 2 9 be fore the co j u ctio i R A w ith the ce t re f t he shado w m m I 2 9 the m oo m ove d 1 7 i R A d i decli atio d there fore the coor di ates f t he m oo at t he m i ddle o f t he eclipse w ere as follo w s ’

To fin d

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on

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decl 14 li e j oi i g the ce t re

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the ear t h t o the ce tre .

n

of

the moo

n

C H A PT ER

XV I I

I

E C L PSES OF T H E

.

S UN

.

I tr d uct ry th O a gle ub te ded at t h c tre f t h earth b y t h ce tr f t h a t t h c m me c me t d th m lar ecl ip e f a E leme t ry th e ry f lar ecl i p e Cl e t appr ach f ear a O d m d Cal culati f t h Be l ia leme t f a partial cl ip e f o

n

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es

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an

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or

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l icati f t h Be el ia eleme t ecl ip e f a gi ve tati

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on

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s

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ss

n

n

n

s

cal culati

to the

on

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s

tr o d u c to r y I f t h e orbit f the m oo w ere i t he plan e o f the ecliptic t here would be ecli pse o f the a t every w m oo A however the orbit f t he moo is i cli e d to the ecli pt ic at a gle o f about five d egrees i t is plai that at the ti me o f e w m oo the m oo w ill ge erally be t oo m uch above or belo w the s to m ake ecli pse possible B t w he the m oo is ear a o d e f its orbit about the ti m e f e w m oo the eclipse f the s m y be e pec t e d W e have already m e t io ed i g 5 8 that 8 t he m oo s asce di g ode m oves b ackwards alo g t he eclipt ic u der t he I i flue ce f uta t i abou t 1 8} year or m ore accura t ely d o d ays 58 m akes a co m plete circui t o f the ecli ptic accou t f t his m ove m e t t he s i its a ppare t m otio passes through the asce di g o d e o f the m oo s orbi t t i tervals f We thus fi d t hat 19 co mple t e re olutio s f the 34 6 62 d ays w ith res pect t o 83 are per form e d i 65 8 5 8 d ays T h e l ti o u or average i terval be t wee two successive n e w m oo s is 2 95 30 6 d ays so that 2 2 3 lu a t io s a m ou t to 65 8 5 3 d ays Th e close f 2 2 3 lu a tio s a d 1 9 revoluti o s ppro xi m a t io i the d uratio 12 0

In

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35 9

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the w ith respec t to 88 is o t a lit t le re markable T hey each d i ffer fro m a perio d f 1 8 years d 1 1 d ays by o m ore tha hal f a day T his curious perio d k o w as the S aros is f m uch Sig ifica ce i co ec t io wi t h solar eclipses S u ppose that at a certai e p ch t h m oo is e w whe t he s is a t 88 d ecli pse f t he t here fore t akes place A fter the lapse f a S aros the s h per for m ed just 1 9 revolutio s w ith regard t o 88 d there fore the s is agai at 88 B t w e w because a also fi d t ha t t h e m oo i agai i tegral u mber d co seque tly t he S aros f lu a t i n s ( 2 23) are co t ai e d i the co di t io s u der which ecli pse is pro duce d will have recurre d O f course t he sa m e would be true with regar d to the m oo de d esce d i g We have see T h S aros is rela t ed to lu ar ecli pses also i C ha pter XV I tha t t here is ecli pse f the m oo when at the ti m e f full m o the is su fficie t ly ear e f the m oo s o des T hu we perceive t ha t ecli pse f the m oo will a ft er t he la pse f a S aros be ge erally followed by a o t her eclipse o f the m oo tha t every ecli pse f either ki d will ge erally be followe d by a ther ecli pse o f the sa m e ki d a ft er i terval o f bo t 1 8 years d 1 1 d ays For i s t a ce there were eclipses i 1 8 90 o J u e 1 6 ( s ) N 2 5 ( m oo ) i accordi gly d D 11 ( d 1 90 8 there ) are eclip es J u e 2 8 ( ) D 7 ( m oo n ) d D 2 2 ( s ) A a other u m erical fa t co ec t ed w i t h the m t io f t he m oo i t Shoul d be o t e d t ha t 235 lu atio s m ake 69 39 0 9 d ays w hile 1 9 year f 365 2 5 d ays a m ou t t o 6939 7 5 d ays T hus w e have t h cycle f M eto co sis t i g f 1 9 years which is early i de tical wi t h 2 3 5 lu atio s H e e we m y ge erally a ffi r m t ha t 1 9 years a fter w m oo we hall have a o t her e w m oo e g 1 8 90 J uly 1 7 a d 1 90 9 J uly 1 7 Whe ecli pse f the s is the poi t o f co mm e ci g or e di g t he circu l ar di c f the m oo as projecte d o the celestial f t he observer is i e x t e al co tact S phere fro m t he posi t io w ith t he proj ec t e d d isc o f t he I t is evide t that a t t his f the observer m om e t a pla e t hrough the posi t io d t he a ppare t poi t f co tact but w hich does t o u t either o f the discs m ust be a co mm o t a ge t pla e t o the s pherical sur faces f t h s It is also obvious that the lin e d m oo of

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j oi i g t he actual poi ts o f co tact f this ta ge t pla e w i t h t he t wo s pheres m ust pass through the positio o f t he observer f i f it di d t d so t he tw o bo dies w oul d t a ppear to h i m to be i co tact Th geom e t rical co ditio s i volved i s lar ecli pses are t here fore a alogous to those w hich we have already h a d to o sid er i C hapter XIV whe discussi g the tra i t f V e us Whe the partial ph a se f a solar ecli pse is abou t t o comm e ce or abou t to d t he observer m ust t here fore occu py a positio the s u r face f that co mm o ta ge t co e to the u d m oo which h s its verte betwee t he tw o bodies i the parallel ca e f t he lu ar ecli pse § 1 1 5 T his c e is k ow s t he t ge t co e to the d m oo m b a the other co mm o s e p w hich t he ver t e x d the are opposite si des f the i m oo bei g t er m e d the mbr a Th e observer w ho sees t he begi i g or e d f a total ecli pse or a ular ecli pse m ust be Situa t e d o the u mbra I t he for m er t he m oo w ill c m I n the latter a m argi o f t he pl e t e ly hide t he d isc f the s brillia t disc o f the is visible rou d the d ark circular for m o f the m oo n

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d I i t s least value 4 W e thus fi d as value 1 the superior eclipt ic li m it T h i f e i o e c lip ti c li mi t is fou d by taki g the lo w e t possible a d d e a m ely respectively values f I f t he geoce tric la t i t u d e f the m oo at co ju ctio be less tha t he ecli pse o f t he s I5 S 1 539 fro m so m e t errestrial sta t io s m ust t ake place abou t t h e t i m e o f tha t o ju ctio Th m a i m u m value f t h i cli atio o f the orbit to t he ecliptic is 5 1 8 6 I f t he values 1 d m oo 5 1 8 6 be subs t itute d fo B d I i the for m ula ( i ) we fi d = T hus we see tha t whe ever at t he t i m e f e w m oo the su s lo gi t u de is w i t hi o d e t he eclipse o f t he m ust t ake place at t hat co j u ctio o f t he s T h e i ferior ecli ptic li m i t is t here f re Fi ally we see that i f B 1 2 4 7 the ecli pse m ust happe If B 1 the there ca ot be ecli pse I f 1 B 1 t he ecli pse m y happe or it m y o t T decide the ques t io w m us t calcula t e d ther e w ill be eclipse or n o t accor di g as B is less or greater t ha the qua t i ty so fou d E 1 If i F i g 90 S M i t h p rp d ic l ar fr m S MN dS M th h rt e t d i t c b etwee t h ce tre f t h h w th at d m app i m t ly °

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on

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an

oon

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o

a e

rox

M M = 2 mB s i n I ’

at ll ite rev lve tim e th d reg rede 9 very rev l uti fa t d it th at ll ite mak r u d i t pri m ary pr ve that there ca t b fewer de th a t h i teger ex t le lar ecl i p e at th a ti Ex 2 .

n s

i feri r ecl i ptic l im it

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If t h e

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i l g itu d th e veme t f t h A i th at f th m d — A9/2 th at f t h i d T h d u rati f a l u ati 2 /( l )A d t h ti m e tak e b y t h t pa fr m t h d i ta c de t a d i t a c e ide f t h th th er i d th um b er f e ti re l u ati c tai ed i thi g ive t h requi red a w r E th m c j u cti 3 A t a cer tai f th d m ju t e ib le partial cl ip e at y p i t f t h b t th ere i gra e t h earth urface P r ve th at Le t A b e t h e d

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1 2 3—12 4]

3 67

T H E SU N

E C LIPSES O F

a gular radii f t h d m d th i r 8 d 8 th e i r d ec l i ati at t h i ta t f c j u cti i i d d e cl i a ti d a 8m th i r h url y m ti [ C ll E xam Fi d t h m i im um d i ta ce b etwee t h ce tr which at t h tim e t appr xi matel y t h quare r t f

where o parall ax

e s,

an

d

an

r

r

D

an

,

the

are

n

m

ons

e

R A

n

an

su n

ons

o

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on

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{ 8m

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R A .

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es

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o

(8m

8, + t

e

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co s

8m

co s

r ve that there m re lar ecl ip e tha lu ar ecl ip e t h average b t th a t t h m fa ce i d i mm d b y t h p u mb ra th u gh t e ce ar i l y e cl i p d rath er m r freque t l y th a t h i e cl i p ed [ C ll E xam ] l u ar ecl ip e wi ll b E 5 P r ve th at at t t i l t ti gi lar m re freque t th a d h ri t al parallaxe f th E Th d em i d i am eter 6 m rb it t fi d t h m ax i m u m i c l i ati f th b i g k w m lar cl ip e every m th t h ec l i ptic which w l d e u re a [ C ll E xam ] Ex 4 .

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ollo w i g m e t hod o f co mputi g t he circu m sta ces o f eclipse f the a t a give terres t rial s t atio is tha t o w ge erally e mploye d I t is d e t o B e sel f h t h ce tre t he earth a li Th n e is su posed to be g p draw parallel to the li e j oi i g the ce tres f t he d m m o e t W e shall regar d this as the a is o f d m oo n a t y the pla e or mal thereto through t he earth s ce tre is k o w as the fu dam e tal pla e T h e posi t ive si de f the pla e f is that w hi h t he d m on are S i t ua t ed is that w hich co t ai s t he a is o f the ear t h d T h e pla e o f Th positive i d e o f the ax is o f is that w hich co tai s the oi t f the equator w hich the earth s ro t atio is carryi g f ro m the p osi t ive si d e f to t he ega t ive si d e T his cri t erio ever p beco m e ambig ous because the pla e O f ever coi cide w i t h the equator d z d T h pla e o f g is per pe dicular to the pl a es f x the posi t ive si de f g is t ha t w hich co tai s the earth s or t h pol e ot beco m e am big uous T his als ca d d ecli atio o f the celestial L t a d be the righ t asce sio The f

n

n

su n

o

n

n

n

e

s

n

n

n

n

n

o

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o

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n

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u

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z ca n

can

n

n

n

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e

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an

O

an

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nn

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e

n

n

.

n

,

Se e

n

al s o

Ch a u

’

ve n e t s

n

an

P r a c ti ca l

and

n

r

n

Sp h e i c a l As t

r

o n o my

.

.

E CL I P SE S O F

TH E

[ OH

SUN

XV I I

.

which is oi ted at by the ositive irectio the a is d o f p p p d decli a t io T he t he R A f f the poi t s t h e celestial res pec t ively are 90 + a 0 ; Sphere poi ted t o by + x g + d 1 80 + 90 — d ; W e he ce ob t ai the cosi es o f t he a gles betw ee t he poi t 8 d t he t hree poi ts j ust give by t he for m ula ( i ) p 2 8 d thus derive the e pressio s t D

o in

o

z

n

n

.

.

n

an

.

n

n

,

°

°

a

a

,

°

,

n

n

n

an

,

n

x

n

8 si n ( a — a )

co s

=A g

Si n

{ z = A si n {

w here

on

an

a,

.

=A

x

n

n

n

,

o

z

,

.

,

n

n

n

x

8 co s d —

8 si n d c o S( a —

)} d + c o s 8 c o s d c o s ( a — a )}

8 si n

co s

a

are t he coor di ates w i t h respect to t he fu dam en tal g a es o f a bo dy i t he directio 8 d at t he d ista ce A L t a 8 a d a 8 be the R A a d decl o f the ce tres f t h e d t he m oo respec t ively the as the li e j oi i g t hese u n oi ts is arallel to w the e m us t have t h e x coor d i a t es f p p d t he g coor d i ates m us t also be equal w he c e d m oo equal A cos 8 i ( a a) 0 ) A cos 8 i ( A cos 8 i d cos ( A si 8 cos d a) A i 8 cos d A cos 8 s i d cos ( ) 0 Fro m t h e firs t o f t he e t a is fou d This gives two values f a e cee ds the o t her by A S ho w ever the valu e o f w hich f m us t be very early t he right asce sio f the there d oubt as to w hich value o f be is t o be chose T h is sub sti t ute d i t he seco d equatio gives t d d here also there c be am bigui ty as t o w hich o f the t w o values o f d d i ff eri g by 1 8 0 shoul d be chose fo d bei g a d ecli atio m ust lie bet w ee x,

,

n

z

x

n

e

a,

n

n

,

,,

,

,

,

an

n

n

.

.

n

an

s

n

z

an

an

n

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n

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o

n

n

n

o

s n

,

a

,

,

,

,

S n

s

,

,

a,

s n

a,

S n

,

,

an

n

n

,

a,

a

or

n

n o

.

.

n

,

n

su n

o

a

n

n

,

,

n

su n

,

x

a

n

n

,

on e

.

.

n

,

n

o

n

n

n

an

,

ca n

.

an

an

n

n o °

90

n

°

an

d

90

n

r

,

n

n

n

°

the p i t D o f w hich d are the coor di ates is So ear t h e ce tre f the d as at the t i m e f eclipse d 8 are very ear to d 8 res pectively the followi g a ppro i m a t e solutio gives a d d w ith all eedful accuracy d the first equa t io w e write t he s m all a gles If i i stead f their si es d i f we m ake cos 8 = cos 8 d a — w e have A A AS

o n

o

n

su n

a,

n

a

,

an

o

an

,

,

o

n

.

n

n

,

an

a,

a

an

,

an

,

,

a,

a

(a

,

the seco d equa t io w e m ake cos ( ) u ity d thus substituti g t he s m all a gles their si es we have In

n

,

x

n

n

n

,

a, an

n

n

n

a

an

,

an

,

n

,

n

an

n

a,

n

n

n

d = s,

a

(s

,

an

8

d

cos ( d

an

a,

d d

a

)

each 8 fo r ’

E C L I PS E S

TH

OF

E SU N

[

ista ce d it is f u d tha t f t he com putatio 6 We he ce ob t ain t h is a gle should be

d

or

n

o

an

n

‘

n

PQ

l

= 7 6 7 00

f

L o g R ta n

radius

VII

n

.

Th e

X

.

ecli pses

of

n

CH

.

t an f

OS)

P ( S

R ta n f

=

a

we

Bu t

.

b where b is the radius f t he m oo have ( R M P ) t f w he ce l = M P t f b I f we take as is m os t co ve ie t the earth s equat rial ra dius a s t he u it o f dista ce f t h m ea ure d 0 2 725 bei g the m oo s hori t al paralla m e n t f l t he t he atio f t he m oo s radius to t he ea t h s equa t orial radius w e have l t cosec 02 72 5 f For e a mple I the a ular ecli pse o f M arch 5 1 905 we have L g R = 99 966 w he ce L g ta f = 9 9 966 7 6 7 39 Th e m oo s hori o tal paralla t his occasio is 5 4 d wi t h the value f f j ust fou d we obtai a

an

an

n

’

n

.

n

o

’

n

o

n

7r,

n

o

r

n

o

,

’

n

n

n

or

n

,

s

e

x, a n

zo n

’

’

r

an

x

n

.

o

n n

n

,

o

n

’

n

.

n

z

,

,

x on

o

’

n

n

an

n

l = 5 72 8 ‘

.

these qua ti t ies i x g L g ta f L g i d g d it w ill be L g cos d p are k ow a s the B esselia n ele m e t s observed t ha t they rela t e t o the wh le earth ra t her t ha to par t icular t t io s thereo e t part f t h calcula t io hows ho w t he B esselia Th ele m e t s are to be applie d t o deter m i e t he circu m t a ces f ecli pse at y par t icular s t a t io All

n

o

,

,

n

,

v z

.

x

,

,

’

’

,

o

,

n

n

n

,

n

n

a

n

e

x

o

e

n

n

an

t

an

n

S

n

t

A ppl i c a i o n

.

,

.

n

12 5

s n

n

o

s

o

,

Of

s

n

an

o

.

th e B e ss e l i

ele

an

m e n ts t o t h e

c al cu

t ti o n ecli pse are prese te d w he t he T h cri t ical phe o m e a f observer is o the pe u m bra or the u mbra I n t he for mer case t he co t ac t f the li m bs o f t he s d m oo is e ter al d t he par t ial eclipse is j us t begi n i g or e di g I t he case f a to t al ecli pse the phase k ow as to t ali t y is just co mm e ci g or e di g w he the observer is o the u m br a ular I the case o f eclipse t he firs t or seco d i ter al co tac t takes place whe t he bserver occu pies t his positio We shall e w s t u dy the case o f the co m m e ce m e t or e di g o f ecli pse It has bee alrea dy stated tha t is the westerly hour a gle

la ion

o

f

an

e cli

ps e f o r

a

n

n

e

n

o

s a

.

an

.

u n

n

n

n

n

n

an

n

n

n

n

a

n

O

n

n

n

n

n

n

an

n

o

”

“

n

n

x

n

.

n

n

n

n

n

o

n

gi v e n

n

n

an

n

nn n

n

n

.

an

.

u.

,

n

1 2 4— 1 2 5 ]

37 1

E CL I P SE S O F TH E SU N

r m G ree wich f t he poi t D d t here f re wi t h res pect t o the observer s s t atio K w hi h has easterly lo gitu d e A the w es t hour a gle f D is + A T h e geoce t ric e ith f the bserver has there fore a righ t asce si w here A d a d ecli a t io p is the geoce tric la t i t u de f K I f there fore p be t he d ista ce f K fro m t he ea t h ce t re d f n t t he coor di ates o f K w i t h re pect to t he fu da m e t al a es we have 3 5 p os i cos i cos d cos d 7 7 ( d a 0} t { p g; p { i d) i d cos gb cos d s ( p A)} T h e values f g d n are to be calculated d 7 d also f E f the particular locality d f t he a m e e poch T w hich was sed i calculati g d g H e ce a t the t i m e T + t w h ere t is e presse d i m i u t es o f m ea t i m e d is u derstood to be a f ourse i t w ill be i f T be pro perly hose ) s m all qua tity ( t h e values f g d ; bec m e f + f t d n + n t res pectively W e have w t o fi d 5 d t hat is t o say the rate p m i n ute t which if d 7 are cha gi g about the t i m e w he the eclipse is visible a t t he place i questio A d f t hese t he three firs t are O b 5 d ) d e pe d p g fixe d f a give locali t y d he e the cha ges i f d 7 a t a give place o ly arise t hrough cha ges f d or u or bo t h A s t o d i t i ery early t h d ecli ati o f t h s d t his at m ost ly cha ges at t he rate f a seco d f e p m i u t e T h ha ges f E d 7 which w o cer us are d e to the ha ges i a T his is very early t he w e t hou r a gle o f the su at G ree wi h m i u t e o f m ea ti m e is d i t s varia t io i about m i u t e f sidereal t i m e or e xpressed i radia s f

o

o

n

n

an

o

’

n

c

o

n

u.

an

n

on

a

s

s

s n c

7

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o

,

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an

7

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n

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o

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n

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n

7

n

n

n

s v

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on

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n

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n

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n

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n

c

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ar

s

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n

n

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e

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n

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n

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s n

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n

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i ere tiati g the e pre sio s f f d 7 w ith regar d to t he we t i m e d re prese t i g the di ff ere tial coe ffi cie ts by f d have cos 2 2 cos A g (p )/ 9 p p ii D ff

n

x

n

an

n

n

s

or

an

7

n

n

n

” ‘

’

f

5 si n

an

'

(

o

’

.

( )

d/2 2 9 2 °

is t a ce f the bserver fr m the is f t he pe u m bra is which brevi t y is re rese te d by I t is obvious L f l t 4 f p t ha t a m all cha ge i C bei g m ul t iplie d as it is by the m all qua ti ty t a f is i se sible d co seque t ly we have as t he Th e d ”

an

n

o

n

n

,

n

n

o

n

or

,

s

n

ax

o

o

n

.

s

n

,

,

an

n

n

n

2 4— 2

u dam e tal equatio f the d eterm i a t io or e di g f t he partial ecli pse

f

n

n

n

or

n

n

n

[

SU N

TH E

EC L IPS E S O F

n

o

XV II

.

the co mm e ce m e t

f

n

n

o

w ( {

E) l

’

’

t (w

E)

{(y

’

v

t (3 /

)

’

v

s lutio f thi equa t io is eff ec t ed as follows We m ake the subs t itu t io s

The

OH

n

o

o

n

s

)l

L

’

’

(i i i )

.

n

m si n M = x m

w hich

M =g

co s

n ;

-

=x — ’

n si n

N

n co s

N =g —n

f

’

’

’

are four au iliary qua t ities T his gives M t ( f) ( g 7) fro m which t w values f M d i ffe i g by W e choose that value which shall m ake 1 8 0 are d eter m i e d Al have t he sa m e ig as x Si 5 the cos M m ust have the sa m e sig as g — n d m w ill be the positive square ro t o f

in

m

n

,

,

M

N

,

x

an

n

,

r n

o

o

n

S

n

.

.

l

n

,

an

o

E)

( 22

’

like m a er N is d eter m i ed square root f

In

n

7

°

n

x

nn

n

(y

t hat

SO

shall be the positive

n

o

'

(y

(w

equa t io ( iv) w e have m L N) t + 2 m t cos ( M where a m i ute o f m ea t i m e is as alrea dy sta t e d the u it W e i tro d uce a o t her a gle 39 such that S

ubsti t u t i g i

’

n

n

t he

n

n

2

2

2

u

’

,

n

n

n

n

n

o

ft

.

n

L si n

m si n

d

r

—N M ( )

.

is give o ly by its si e t here is a ch ice f two su pple m e tal a gles f We ch ose t hat w hich lies bet w ee o e d d 90 so that i ositive the 90 d p As

dz

n

n

n

n

°

or

n

r

t + 2 mn t 2

co s

cos

o

o

o

.

°

an

2

n

r

n

s

n

n

(M L

2

m

2

m

2

cos

2

M (

N)

L

2

cos

2

d r

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T H E SU N

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by which we lear the poi t s f the solar disk which the m oo touche at the first d last m m e t s f t he ecli pse f t he ecli pse wi t h greater T d e t er m i e the circu m s t a ce accuracy the cal ula t io Sh ul d be re pea t e d u i g t h e values fou d i g d T i stea d f T accor di g to whether i t is t he begi f T or e d f t he eclipse that is sough t an

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E C L I P SE S O F

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[ Math

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C H A PT E R

X V I II

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I

M OON

O CCU LTA T O N S O F STA R S B Y T H E

i ve ti gati o

Th e

12 6

.

n

s

of an

n

o

ccul tati

.

on

t i ga t i o n o f a n o c c u lt a t i o n It occasion ally happe s that the m oon i the course f i t s d a star T his ph m ove m e t passes bet w ee the bserver calle d A the s t ar m y f this me i o cc l t ti o m ur ose be regar d e d as a m athe atical poi t the e ti c t io f p p the star by the m oo s adva ci g li mb is usually n i sta ta eous n he o m e o though occasi ally owi g d oubtless to the m argi al p irreg laritie o f the m oo s li mb the phe o m e o is o t qui t e SO si m ple Th e reappeara ce f the s t ar whe t he m oo has j us t assed cross i t m also be bserve d though i this case the p y bserver should be fore w ar e d as to the precise poi t the m oo s li mb w here t he star w ill sudde l y e m erge It is ea y to see the astro o m ical sig ifica ce f t he observatio n f occultatio T h e t i m e o f its occurre ce d epe ds b t h the f the ob erver d the positio Th m ove m e t o f the m oo f d lace the star bei g k o w with all esirable recisio o n p p accurate bserva t io o f the m o m e t o f the star s disappeara ce d t he posi t io gives a rela t io be tw ee the place o f t he m o the bserver T h e bservatio m y be available f of ccurate deter m i a t io o f the place f the m oo or it m y be se d f fi di g the lo gi tude o f the obser er i f co mpared w i t h the si m ilar observa t io a t a other sta t io f k ow lo gitu de Th follo w i g m ethod f calculati g the t i m e at w hich t he d isa ppeara ce or reappeara ce o f a occul t e d s t ar takes place is due to L agra ge a d B essel 12 6

Th e i n

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ves

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O CC

UL TAT IO NS

coordi ates cos 8 s i

Th e

n

x

o

f

t he

OF

S TA

RS

MO O N

TH E

BY

[O H

X V I II

oo re ferred to t he sa m e a es are cosec m ; g cos 8 cos cosec i 8 cosec m

n

x

'

n a

.

a

7r ,

’

s n

z

be the ra t io w hich the dista ce f the m oo fro m the observer bears t o its dis t a ce fro m t h ce tre o f the earth then will be the d is t a ce f the m oo fro m the place o f A cosec d t h e proj ectio n s o f this d ist ce observatio the t hree a es respecti ely are cosec A cos 8 cos A cos 8 s i cosec A S i 8 cosec w he n ce w e ob t ai A cos 8 i cos 8 si c sec c sec cos i ) S d p A cos 8 cos cos 8 cos cosec cosec cos S pc 8 cosec A i 8 cosec i si p w hich m a y be cha ged i to i cos 8 si cos i A cos 8 s i s S ) p d A cos 8 o cos 8 cos p cos 3 i n cos S If A

n

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n

n ,

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by A d eli m i ati g e pressio s jus t give we have A si 2 s i 9 cos 8 si ( a A ) si n (S si n cos d p A) A si 2 0 3 9 i 8 c o s D — cos 8 i D o s ( a D i cos D cos si cos S i c ( p p { p A cos 2 D + c s 8 6 0 8 D cos ( — A ) Si n 8 si D cos cos cos i n i si D S s b ( p t g { n

,

s n

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by the

5 12 6]

OC C

ULTAT I O N S

ST A

OF

RS

B Y TH E

M O ON

hese for m ulae e able 2 d 9 to be d eter mi ed d are S ecially d a te d f the stu d y f occul t ati because at the b p p gi i g or the e d f occul t atio t he star is th m oo s li mb d we have 2 A the si e f t he m oo s se m i d ia m e t er varies i versely as i t dista ce we m ust have A i si d t here fore A i 2 si I t ro duci g thi i t o the equatio s ( ii ) we obtai the followi g re m arkabl e for m ulae t rue a t t h e m o m e t s occulted star f disa ppeara ce or rea ppeara ce f T

n

a

an

or

nn n

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9

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cos p d si n (S A ) si c o s 9 = si 8c sD cos 8 i D cos ( A ) D Si i cos cos i D cos S A ) c ( )] d { p p It is plai t hat a d are co ected by t he c sta t relatio i r radius f t h/ a di s o f m oo Th e ra t io f the m oo s ra dius t that f t he earth is ter m ed l a d is equal to 0 2 7 2 5 T hus we have fro m ( iii) h si 9 cos 8 si ( A ) cosec A) p cos ps i ( S 8 si 8 cos D cos si D cos cosec 1 cos 9 A ( ) { } i D i 0 cos cos cos s A S ( { )} p p 3 Fi ally squari g d ad d i g we ob t ai t he f llowi g fu da w hich co tai t he theory o f t he ti m e f m m e n tal equa t io occ ltatio m e c e m t or e di g f cos A cosec i A 8 si a S ) ) ( p )} ( d { i i cos D cos D 8 8 s ( A ) } cosec [{ i i cos D cos s D A S ) }l ( p{ 3 t he t he o ly I f the coordi ates f the observer be give u k o wn i this equa t io is S t he ti m e T h solutio f t his equa t io f S w ill t here fore S how t he m o m e t f t he beg in i g or t he e d f occultatio T h e equa t io n f S is ecessarily a tra sce de tal equa t io f it has to re pr ese t all possi ble occul t a t io s i i fi i t e ti m e T apply it to y par t icular occulta t io we m us t a ppro i m ate m ethods L e t T b e n assu m ed ti m e very n ear the true ti m e T + t at which a certai occultatio takes place t is thus a s mall qua tity s i n 7r ,

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OCC

UL TA TI O N S

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B Y T HE

the ter m s f the equatio c be e pa de d co vergi g series f po w ers o f t W e hall m ake cos 8 si ( a A ) cosec p pt i A cos cos D cos cosec D 8 i 8 ) ( } q { d

an

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[C H in

n

.

XV I II

a rapidly

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S

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n

7r ,

s n

s n

r co s

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a

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qt

qr ,

’

d s i n (S

D cos A cos D cos si S t ( ) } d { d I t is su ppose d that p q v are ca lculate d f the ti m e T a d f m t w are ter s i volvi g which e m y first assu m e the v p g appro i m ate value ero Th e equatio ( v ) the beco m es si n

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accu rate value o f t For a co ve ie t solu t io

n

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these equati s w e m ake — = =m i M ; si p p — =m = M eos v s v ; q q where m M N are four au iliary qua t ities k = (m s i M + i N t ) + ( m o M + n eos N t ) —m i — N M N m cos M t} ) { ( ( ) We w i tro duce a other au iliary qua ti ty d such tha t n

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Si n

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m {

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=

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cos

cos ( M

r

r,

.

N)

cos ( M — N ) 4 h si d W assu m e that d is less t ha d the t he u pper ig correspo ds t o t he d isa ppeara ce o f the star behi d the m oo a d the lo w er to i t s rea eara ce pp n

—m

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m si n ( M

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the d is i m agi ary d there is o occul tatio I n d ra wi g this co clusio i t woul d be pro per t re m e mber that further pp o i m a t io m y be ecessary t o d eci d e w hether t his co ditio is truly satisfied T e a m i e this we t ake f t i t s m ea value m cos ( M N ) / n

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UL TA T I O N S

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[C H

X V I II

.

a gle subte de d at the m oo s ce t re by t h e star d t h f d d m ole at the m o e ts isa eara ce rea eara ce is very p p p p p early The

n

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18 0 — 9 = N i °

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hus the ecessary for m ulae f solvi g the problem f a have bee b t ai ed o ccul t t io For co ve ie t m e t hods f co d ucti g the calculation s re fer w hich tables are give by e ce m y be m a d e to t h e e phe m eris i w hich the w ork is facili t a t e d 2 7 O t 1 90 9 t h d ecl i ati f th m 1 A t m i d i ght i E d d cl i ati f th m th e i crea i g i d th 4 36 very l o b y 2 3 0 d 164 re p ctively S h w that a tar i c j u cti with t h m at m i d i g ht ca t b ccul ted at ab ut t h tim i if th tar decl i ati i le th a 3 it b i g f thi c j u cti h m d h i ve th a t t m f m e i a m e t r d h r i t al arallax t i i p g T

or

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e ce t h i e f t h d i ffere ce b etwe t h m dec l i ati d th tar at t h tim e f c j u cti m u t t ex c ed i c ec (64 0 251= i f th m rb it t t h ecl i ptic b 5 20 2 I f t h i c l i ati E h w th at t h m w ill at me ti m Oth er ccul t y tar wh e la itude rth uth i le th a 6 38 [C ll E xam ] th ecl i ptic wi ll b ccul ted b y t h m S h w th at a tar i E 3 tati d 2 2 ti m at ach pa ag f a t h ear th b etwee 1 7 m t rb it th r ugh t h tar A u me m i d iameter f m f th m d d 14 h r i tal parallax f m b t wee 61 18 d b tw 16 4 6 i cl i ati f rb it t ecl iptic b twee 5 19 d 4 53 [ M ath T rip ] ccul ted V e u I t w ta ted i E 4 O F b 2 9 18 8 4 t h m th at V e u w ul d b t h m er i d i a a t t h Ti m b i g p m th m the th re day l d d that t h ccu l tati w ul d la t ab ut h ur d a quarter Fi d r ugh l y wh it c mm e c ed d h w that t h tat me t t t h d urati f ccu l tati i t i c i t t w ith t h k w a gular d i amet er f t h m [ Math T ri p ] H

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C H A PT E R XI X

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I N V O LV I N G

PR OB L E M S

P

he

n o me n

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T o fin d t h e

r i i g d etti m ea ti m e f r i i

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i i g d tti g f t h mo T w i l i g ht Th u dial f p i t th u urface C rd i ate R o tati f th m S u m er meth d f d t rm i i g t h p iti R

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tt i n g Le t R ( Fig 94 ) be t he a t su rise a the first poi t o f L ibra where ER N is the hori o R t he ecliptic E the equator A E is the eas t erly poi t the equator produce d fo 12 7

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INV O LV I N G S U N

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[C H

XI X

.

the directio fro m A to E w ill reach the m eri d pro d uce d f a fur t her dista ce equal to S the si dereal d ia ti m e wil l reach T the first poi t o f A ries H e ce w e have 90

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ROB L EMS

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= 90 — S , °

lo gitu d e f the u is A d R A = 1 80 — A because A is f the arro w head a is the m easure d fro m T i the directio a i m uth f t he risi g s m easured fro m N t h e orth p i t f t he hori o i the usual direc t io by east d south P is the pole the altitu d e f t h T h d PN e pole above the hori o e p a gle at E be tw ee the equator a d the hori o is equal to e ith to the pole 90 — a as tha t is the d is t a ce fro m the is the i cli a t io o f the ecl i ptic to the hori o 180 — f t he ecli ptic a d w is t h e obliqui t y M a y questio s t hat c a be pro pose d w ith regar d t o the ri i g d setti g f the be solved by the tria gle E R A I f c d that n e f t he other w e assu m e that d d are give elem e ts f the tria gle is k o w t he re m ai i g elem e t s c n be deter m i ed T fi d the t i m e o f su rise or su se t w he the lo gitu d e A is give w e deduce fro m t he for m ula p 3 Th e

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his equatio m y be pu t i t t he for m i w he ce S B A or S 18 0 B A o e o f which corres po ds t o t he i si g d the other to t he setti g I f t here be a s m all cha ge AA i the su s lo gi t u d e A t here w ill be a corresp di g cha ge AS i the si d ereal t im e f su rise or su se t Th rela t io betwee AA d AS is ob t ai e d by d if fe e t i t i g t his equatio w hile regardi g d p as co s t a t w e t hus fi d si A ( s i A cos S cos w cos A si S ) A S s S AA but fro m the t ria gle E R A w e have at o ce cos A si S si n a si A cos S cos w he ce cos S c o Se c A cosec AA AS lso cos S cosec A si s ize s i n

co s

x

,

¢ co s 8 co s 9

,

subtracti g n

cos (pc o s 8 cos ( 9 + 4 ) cos gx 3 2 cos 4) cos 8 i ( 9 cos i ) 4 5 m l t i plyi g t he first by i 4 the d the seco d by cos 4 squari g d a ddi g w e eli mi a t e 9 d ob t ai 2 i < i cos si c z x ( cos 5 1 3 cos 96 cos 8 si cos z

2

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RO B LEMS

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1 31

Th e

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su n

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IN

V O L V IN G

SU N

MO O N

OR

[O H

.

XI X

.

We m y su pp se t hat the place f t he t he celestial 2 4 hour s sphere d e t cha ge a ppreciably i d a pla e t hrough t he earth s a is d t he w ill i t ersec t t he terres t rial equa t r i tw o poi ts w hich m ove u i for mly rou d the equa t or i i co seque ce o f the su a ppare t diur al rota t io I like m a er we e t ha t i f a post were fi ed per pen dicularly i to t he earth a t the orth pole S as to be coi ci de t with the ear t h s a is its hadow woul d m ove u i for m ly rou d the hori z o so that the posi t io f the n d t here fore the appare t tim e w oul d be i dica t ed by t he poi t i w hich the sha d o w crossed a u i for mly grad ated circle with its ce tre i the axis f the d its pla e perpe dicular t o the earth s a xis T hus w e pos t have the con ce pt io o f the s dial A S t h di m e s i o f the ear t h are so i co si d erable i c o m we m y say that i f a t y pariso t t he dis t a ce f the s poi t f the ear t h s sur face a pos t or ty l e as i t is calle d be fi x e d s n arallel to the earth s a is t he shadow f t he style cast b the o p y i its daily m otio o a plan e per pe dicular to t h e t yle w ill m ove rou d u i for m ly d by sui t able gra d ua t io will ho w the appare t tim e Th hour li es the dial are t o be draw a t equal i tervals f T h i cli a t io equals the lati f t he style to the hori t ude d t he i li a t io f the d ial to the hori o is t he colatitude T hus w e have what is k ow as the e q t o i l s dial While the style is always parallel t o t he earth s a is the pla e o f t he dial m y be arra ge d i d i ffere t posi t io s hori o tal ver t ical or o t herwi e Th grad ua t io o f t h d ial is u i for m o ly i the equatorial s d ial d we have w to c sider the gradua tio f the dial w he otherwise d lace so that the sha do w f t he p s tyle shall i dicate the appare t t ime S u ppose tha t t he pole f the F m 96 f la e t he d ial is t t h e poi t p O f t he celes t ial phere f which the or t h polar d ista ce is p d west hour a gle 10 a

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§

1 31]

P

( Fig

RO B L EMS

S UN

I NV O L V ING

OR

MO O N

be the or t h celes t ial pole P Z the m eri dia P P the hour circle co tai i g the s S H the pl e f the d ial T h e poi t S is calle d the s bs tyle d PS 90 p is t he he i g ht h t t e t l T h hour li e corres o d i g the hour circle P P e e f y p is give n by H w here O H T gra duate t he dial we require t o k o w t he arc S H = 9 cor es po di g t each solar hour a gle h P ro d uce H P t o P so that H P = P m ust be a right a gle because OH 90 a d co seque tly cos p t ( h k) t 9 ( i) As p d k are k o w n this equa t io gives the value f 9 = S H f each value f h T m ark the hour li n es articular i stru m e by t p y observa t io we proceed a s follows It is su m e d that t here is ordi ary gr d uatio n fro m 0 to 360 the d ial the ce t re o f graduatio bei g t he poi t i w hich t he s tyle m eets the pla e o f the dial d t h origi fro m which t he a gles are me asured bei g the li e through this poi t d S t he ubs tyle I t is also assu m ed t hat p is a k ow a gle W h e t he n has a k o w hour a gle h le t the observed posi t io o f t he shado w be d we have t 9 cos p t ( h I ) We t hus fi d k d co seque t ly f each value f h we co mpute fro m ( i ) t he correspo di g val ue f 9 T hus the s m f o e t the hour a gle t he or t he m d ial w ill sho w at y a ppare t ti m e d by applica t io o f the equatio f t i m e the m ea n ti m e is ascertai n ed d ial m ost u ually see is t he o called hori T h e for m o f s n dial i n w hich as t he d ial is to be hori o tal 0 m ust on t l coi cid e w i t h the e i t h Z ; we thus have 1 0 d Le t P

96)

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l a s t hour li es t hat ee d be d ra w the d ial are those correspo di g to t he case w here t he s reaches t he hori o whe its d ecli ati is greatest I this case i f h be the hour a gle cos ( 18 0 h ) t n t ( 2 3 Th e

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396

P

RO B LE M S

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M OO N

OR

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.

x 1x

whe the value f h thus obtai ed is substitute d f h i ( i ) w e O btai 9 e tre m e t y pe f s d ial ? is that i w hich t he dial is A n d the s t yle is parallel t o t he m eri d ia xis but arallel to the earth s a the ot i p h la e f t dial e p L t P Z P ( Fig 97 ) be t he dial parallel t o the pla e f t h e m eridia A B is a t hi recta gle s t a di g per pe dicular to the f la e a er w hich t he u er o f the a d p pp p p e dge A B parallel to t he terrestrial a is P P is the style T h e s n i the diur al F 97 m otio m y be su ppose d to be carried by a pla e rotati g u i form ly rou d A B d he ce t he sha do w A B f t h e e dge A B w ill a l w ays be parallel to A B d at let us say t he dista ce W he the n is i the m eridian is i fi ite W he n t he su s hour a gle is 6 the I ge eral i f d be 0 the height o f t he s tyle above t he dial ’

o

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here h is t he su s hour a gle Fro m t his equatio n t he value o f wf each value o f h be fou d E 1 Sh w h w t c truct a d i al f which t h d i al h al l b vertical d faci g d uth d i which t h tyle i d i rected t t h uth p le T hi m y b b t a i ed a partic l ar ca e f t h g eral th e r y b y m aki g I = 0 p = pi qua ti (i ) d irectly f ll w th h ri L t S ( F i g 98 ) b t h p i t d uth N t h adi r P t h uth p le P H th h ur ci rcle c ta i i g t h th e fr m t h t r i a gle N B E w h ave ’

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F IG

98

r ve that y d ial b graduat d b y t h f ll w i g r u l L t T b t h ti m e a t which t h h ad w f t h tyle w r mally the d i c t h t h cel ti al ph ere f pr j cted th rmal t t h d i c the t h mark f ti m e T i i cl i ed t t h mark f ti me T at a gle — t t ( T T )} { [ C ll E xam ] x mpl f thi k i d f di l u t W i mb r M i t r 1 A Ex 2 .

e

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rotatio i tersects t h e su s sur face i a great circle k o wn as t h e P oi t s t he su sur face are said t o have helio to eq so l graphic lati t u d e d lo gitu d e w i t h re fere ce t o t he solar equator T h e heliographic latitu d e o f a sol r poi t S is the perpe dicular d the l gi t u d e O f S is the fro m S to the solar equator a e fro m a s t a dard poi t 0 the solar equa t or t o the foot f the er e dicular p p I Fig 99 ON is t h sectio o f t he sur face o f the s by n

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la e w here the oi which t he O f t he ecli ptic i t i 0 p p s sur face is m e t by the li e fro m t he su s ce tre to P a d su t he lo gitu des i crease i the d irectio show by the arro w he d O N is t he solar eq ator a d N is the asce d i g o d e o f the solar equator t he ecliptic T his poi t re m ai s fi e d i t he la e f the ecli t ic because the su s equator has reco is ble o n p p g f precessio T h e lo gitude H m o t io f N m easure d the ecl i ptic fro m 0 the equi o o f 1 9090 is 7 4 A s t he s is d there for e ca t a Soli d bo dy ot have a per m a en t G ree w ich resor t is m ade to a s pecial m etho d f i dicati g the poi t lo gitudes Th e poi t 0 w hich is a d pt e d as the origi o f s o l f t he solar equa t or 0 is d efi e d to be t h particular poi t w hich happe e d to be passi g throu g h N t G ree w ich m ea oo 18 5 4 B y t he rota t io f the l t J 0 is carrie d t o w ar ds N w i t h a u i for m m otio w hich would bri g it rou d t he circu mfere ce i 2 5 3 8 d ay Th solar equator is i cli e d to t he eclipti at t he a gle 90 — xh = 7 T h e coor di ates 3 7x are the la t itu d e a d lo gitu d e f a poi t P t he su s sur face wi t h respec t to ON d m eas red fro m t he origi 0 I like m a n er h B are t he heliographic coordi ate f P w ith res pect to O N d m easure d fro m t he origi O n

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RO B LE M S

I NV O LV I N G

MOON

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sho w that the egative sig shoul d be attribute d t o si n P it su ffices to take the case o f i G) It is t he obvious that the positio a gle P m ust be but this w oul d o t be the case u less the radical i the e x pressio f i P h d a egative S ign i n P m y be w ri t te i the for m f cos ( C A D h ) w here f is egative qua t ity d where h is i d epe de t o f G) it is easily show tha t P is positive fo hal f t he year ( fro m J uly 7 to egative f the re m ai i g hal f Th m axi m u m 5) d Ja d the m i i mu m is value f P is O ctober 8 A pril 6 o I t i requi red t fi d t h val ue f P J ul y 1 5 th 190 9 fr m th e E 1 f ll w i g data To

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the t erres t rial equa t or d A t he arc f the m oo equa t r de the ear t h s equa t or t o i t s asce di g fro m i t s asce di g od e t he ecli pt ic 83 is as u ual t he lo gi t ude f t he a ce di g the eclipt ic d e f t he m oo s orbi t

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be fou d wi t hout a mbigui t y d t he last three i like m a er give d 83 d t he c i ci d e ce f the t wo values f : thus i d epe de tly fou d provide a use ful check o the accuracy o f the work f the m oo coi ci d es wi t h t ha t f A t he period o f ro t a t io its rev lu t i rou d t he ear t h t he m oo kee ps early t he sa me face to the ear t h O wi g however to the i cli a t io o f the d t o other circu m sta ce c o m oo s equator t o the ecli ptic a certai m argi rou d t h e m oo li m b t d w ith i t s m oti occ sio ally passes out f vie w d a corres po di g m argi o other side co m es i t o view T his phe o m e o is k ow as th the li b ti o f the m o a ce d i g d e 1 E O S ept 2 8 h 1 908 t h l g itu de f t h m f th m eq uat r t t h earth d et r m i e t h i cl i ati i 70 de f t h m equat r t h equat r t h t rre f th a c di g t rial equat r d t h fr m t h a c d i g de t h ear th eq uat r t t h e cl i ptic de di g t hé a c W b tai fr m t h ab ve f r m ulae n

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cyli drical socket H H i the base LM It m us t be u derstood t ha t axis I m y be hori o t al or ver t ical or i y t her posi t io bu t its direc t i is fixed rela t ively t the base L M d f o u rs e i t has freedo m fo o her o t io t ha rotatio t m y Th e beari gs at H are fi e d t A B d carry C D k ow a is II which c ro t a t e freely i i t s beari gs though lo gitu di al m o t io t hrough i t beari g is A s A B is r t a t ed t presu m e d C D is carrie d with it d A B is fi e d d t h e a gle be t wee C D X Y i the d ia m e t er o f a grad ua t ed circle fi e d rigi dly to LM d f which t he pla e is per pe d icular to A B f Th e gradua t io this ci rcle is t o be fro m 0 t o ole o f this ci rcle is t o b e Th n

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4 34

is f

T H E G ENE

RAL I Z ED

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UM ENT

[O H

xx i

.

the ce tre f t he ci cle X Y t o the n ole O f that circle the a xis T h a is f C D is f o m t he ce tre f X Y to the ole o f X Y t he telesco pe is f o m the eye piece t o the bject glass a n d the r a d ii f the circles are f o m t heir respective ce t res t o t he circu m fere ce Th e poi ters are as has bee said rigi d ly attache d t o A B If we thi k f t he poi ter as a straight li e rigidly a t t ched t o A B d perpe dicular t hereto this lin e will be parallel t o so m e ra dius A the a xis A B is tur ed r u d 360 t his o f the gra d ua t ed circle parallel ra d ius w il l als m ove co m pletely rou d the circu m fere ce arro w head the poi ter to i dicate its se se the I f t here be we d efi itely ssu m e t he re d i g to be t ha t i dicate d by t h e rad ius d aw fr m t he ce tre o f t he circle parallel to t he poi t er t he d irec t io i dic t ed by its arro w hea d a d i I like m a er a poi t er f t he circle X Y m ust be fi e d d be perpe dicular to a is II as the i gidly to a is I S f geo m e t rical t heory is co cer e d we m y m ake the sa m e p i t er W e have o ly to i m agi e the poi t er as the d fo both circles co mm o pe pe dicular t o A B d C D d rigi dly fi xe d to A B d the T he t his l i n e will be parallel to the pla es f bo t h circles radii i each circle parallel t o this li e w ill give the corresp di g rea di gs f each circle L t R be t h gra d uatio i circ l e I ( i e X Y) i d icate d by t h ra dius o f t hat circle parallel to the poi ter just d escribe d d i the se se show by the arro w hea d o t he poi t er L t R be t he graduatio i circle II ( i s X Y i dicate d by the ra dius o f tha t circle parallel to the poi ter d i the se se show by the arrow head the poi ter T he w hatever p i ters be ctua l ly sed provide d o ly t ha t t hey are fi e d to a is I their i dica t io s o ly be R + A R d A R are cer t ai i de d R + A B respectively where A R the i s t ru m e t It will duly appe r e o s which are co sta t f later how t he qua tities A R d A R are to be deter m i ed We shall first i vestiga t e t he rela t io s between R d R d t he coordi ates f t he b o dy the celestial s phere ro

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t o n th e s ph e r e We shall w s t u dy t he ge erali ed i stru m e t by the hel p o f n es i oi s the celes t ial s here corres o i g to the li the d t p p p ge erali ed i stru m e t po i n

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1 4 0— 1 4 1]

T PI E G E NE

R AL I ZE D

I NST

RUM E N T

4 35

raw fro m y fi i t e poi t 0 li es parallel t o t he li es f the ge erali ed i stru m e t as already e plai ed each i the se se f t he arrow hea d t h e corres po di g li e E a h ra dius f the celes t ial phere su pp se d so d raw will ter m i a t e i oi t p the sphere d t he betwee y t wo such poi ts will be equal to the a gle betwee the two cor es po di g li e f t he i stru m e t f t he celes t ial body D ra w also fro m O a li e i the d irectio T hi li e will su ppose d t o be a s t ar w hich is u der bserva t i be c i ci de t with t he li e draw t hrou g h 0 parallel to t he a is f t he t ele co pe whe t h telesco pe is directed u po the sa m e star L e t this poi t be S ( see Fig i like m a er le t B be t he oi t corres o d i g to a is I D t o a is II d V t o the co m m o p p t he two circles poi ter f D

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1 14

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be the polar circle f B S that N V is the grea t circle N V will re prese t i g the pla e o f ircle I A y t w o poi ts there fore be eparated by equal t o t he a gle be t wee t he two corres po di g radii f X Y A B is the ole f t he circle i creases fro m N to V ( as show by t he N V t he gra d ua t io arro w hea d) We have already se t tled t ha t the gradua t io at t h e Le t N V n

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2 8— 2 o

THE

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Z ED

IN S T

RUME NT

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OH

xxI

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B A s t he circle X Y re m ai s i oi t is to be t he sa e V m p m t o i t io ho w ever t he i s ru e t be ro t ate d abou t or we A B D C p f t he i stru m e t are co cer e d a s su h m ove m e t s m y so f reg rd N V as a fi e d circle the celestial sphere n

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T HE G E N E

cos H S cos H N = cos R -

co s

HS

Si n

[OIL

x x1

H V, H V + si n R

co s

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cos

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1 4 2 g ]

R A LIZ ED

T H E G ENE

I N ST

RUM ENT

4 39

d 8 c require d qua tities b e calculate d by these fo m ulae fro m the O bserve d qua t ities R d R it bei g assu m ed t ha t t he co s t a t s f the i s t ru m e t i 9 X g are k n o w E 1 m f th Th f t h th ree l eft h d m em b r f th qu are equati ( 3) f t h g eral i d i t rum e t i qual t u ity V er i fy th at t h ame i t rue f t h m f th f t h th r e r i ght h a d quare m m b er E 2 D ter m i e wh at t h equati f th ge ral i ed i t r me t b c m e wh ax i I i perpe d icular t ax i II (g = 0 ) whe there i rr r f c ll i mati i t h t ele c pe ( = 0 ) d wh e 8 t h c rd i at f th le f ci r cle I ly i trum e tal c ta t i t h xpr i th It i b vi u that X= 90 + whe ce l im i ati g X d 9 d d maki g g = = o t h qu ati (3 ) b c m e Th e

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T H E G EN E

Ex 7 .

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u le n

I NST

RUMENT

[O H

.

XXI

iametricall y pp ite p i t t h cele ti al t h r ad i g f ci r c le I I w h t h g eral i ed i t r ume t t b d i rected t P Sh w th at t h e i t r u me t ca

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1 43

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th t w po ib le l uti i g b t s l i b etwee 49 0 d — 90 w ca d ter i e which ai r r t i t b take E S h w th at f every real p i t t h e cele tial Sph ere except t h 6 l d t i l f ci rcle I t h c rre p d i g readi gs R d R will b e ither b th real ex cepted c e R i b th i mag i ary d th at i th i det rmi ate We h ave E X 3 a

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h w th at i ge eral t h t l c p c ul d o t b d irected t t h le f ci rcle I u l t h read i g that ci rcle i dicat d ther f t h i mag i a y ci rcu lar p i t at i fi ity ed th at t h c rdi at f t h le f 14 2 E 2 w h ave m e ti I = X— 90 i u b tituti th e e ci rcle I i ve b d d 9 g g y — =O M w d N= X th a t 8 M= i 9 i 8 8 i (X— ) 9 ( ) d N 0 T ati fy u der th e c d iti t h equati Ex 8 .

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o

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TH E

G E NE

RA L I Z ED

I N ST

RUME NT

[O H

XX I

.

we btai a bigui t y m d 8 fr m t he three equatio s w ithout y T his is t h e d irec t proble m w hich has t here fore always solu t io d ly e B t i the i verse proble m d 8 are give d we are t o seek R d R fr m t he equatio s ( iv ) ( v) ( vi ) 1 4 3 T here are t w o solutio s real i m agi ary or c i ci de t to th i s i verse problem so tha t i f the ge erali e d i s t ru m e t be poi te d a star i e way it i ge eral be also poi t e d the sa m e s t r i quite a di ffere t way It m a y t be possible to direc t the i t ru m e t t he star by y real setti g bu t i f it is there are ge erally t wo to t ally di ffere t d ispositio s f the i s t ru m e t by which t h e star be bserved T here are thus two d i ff ere t e d R which eq ally corre p d to pairs f values fo R d 8 pair f values f Fro m t he equatio ( iv) 1 4 3 w e c d eter m i e si R a d i f t his is j l we sa t is fy ( iv) by ei ther f t h t w real a gles R d 18 0 R I trod uci g the firs t f t hese u pple m e tary values i t o ( V) a d t aki g t h is i co j u c t io with ( vi ) we have t wo li ear equa t io s fro m which both s i R d R are d e t er mi ed d thus R is k w w i t hout a m bigui t y a s to i t s qu d ra t Th e value f R so f u d we hall ter m R Whe 18 0 — R is subs t itu t e d i ( v ) the equa t i so btai e d i f take i co ju ctio w ith ( vi ) will i like m a er give a o t her T hus f value f R which we Shall ter m R 14 3 E give values f 8 w e have t w o solutio s i R R n d 18 0 R R d we lear tha t i f t here is there are ge erally two e t he d i ff ere t positio s i whic h the ge erali e d i s t ru m e t c be a give s t ar O e f these is called the righ t poi te d u po f cha gi g the d the o t her t he le ft d the pera t io positi i stru m e t fro m o t he other is ca lle d f these positio s t O

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r co s

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xpre i

ss o n

R + co s r

sin

R cos R

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r cos

R si n R

'

ch a ge i f t h e ge eral i ed i tru me t b r verse d d red i rected t ame p i t f t h e cele tial ph re a d explai t h e ge metri al m ea i g f fact W se e fr m ( ii ) p 4 4 1 that t h g i ve ex pre i i equal t o

s n ot s

n

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n

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8 s i n 9 — s i n (X

ss o n

n

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) co s 8 co s

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c

n

n

O

TH E G E N E

RALI Z ED

I NST

RUM ENT

[

C H XXI .

144 i R is alrea dy how n i t cha ged by reve sa l i e the two values f R i a per fec t ly a djusted i stru m e n t w oul d be su pple m e tal H e ce

AS

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S

n

O

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s n

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R1 + A A

A

90

R g( ;

°

.

hus fro m a si gle pair f righ t d le ft readi gs a d is t a t objec t w e deter m i e A I f the d ista t m ark be a s t ar it is to be o t e d t hat t he d iur a l m ove m e t o f t he heave s w ill i certai cases m ake t he ord i ates f the s t ar di ff ere t i the seco d O bservatio from wha t they w ere i n the firs t Th followi g procedure w ill ge erally su ffi ce t o re m ove t his di fficulty T here are t o be t w o bservatio s f the star i the right d o e observa t io f h ositio R t s t ar the le t m s e i f ; p m w m osi t io n a t a o e t hich is m i d way be tw ee the t wo righ t p Th e m ea O f the t wo for m er is to be take fo R O bserva t io s eli m i a t e the e ffect O f the diur al m otio f m os t T hus w e c ractical ur oses p p p f t hi particular i stru m e tal co s t a t is T h e d eter m i a t io so sim ple that i w hat follo w we shall al w ays presu m e t hat t h correc t ion has bee m ade so that the R f r for m ulae is i deed the arc K V f Fig 1 14 T h i de er or o f circle I or X Y d m 1 1 Fig ca o t be eter i ed u t il certai other co sta ts 3 ) ( i ves tigate d e c t e d w i t h the i s t ru m e t have bee T

n

O

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n

n

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n

n

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n

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.

t e r m i n a ti o n o f q a n d b y O b s e r v a t i o n s o f o f th e i s tr u m e n t s t a r s i n b o th r i gh t a n d l e f t p o i t i o n d R be the readi gs o f circle I i the right d le ft Le t R m f d d ositio s t he i s t ru e t whe irected to the sa m e ista t p m ark it bei g u d er t d tha t i f t he m ark is a s t ar the e ffec t f m m m a are t ove e t is to be eli i ate t he way alrea y i d d y pp e plai ed It w ill be show n that t h i de err r o f circle I h s e ffect o the fi di g f g d by the prese t process d t here fore we m y regar d it as ero while the i d e error f circle II w e have already correc t e d W e hall w wri t e t he i 8 1 42 ) f both the right d le ft positio s f r m ula f W e have f t h righ t positio 14 6

Th e de

.

r

s

s

an

1

n

2

n

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a

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or

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x

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a

n

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or

s n

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o

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n

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5 —1 4 6]

TH E

-

G EN E

RAL I Z E D

I N ST

8 = — c o s 9 si n g s i n

si n

RUM ENT

r

cos q si cos R cos 9 cos q cos i R i 9 cos Si R cos R i 9 s i g cos c R i R the le ft positio d f in 8 = cos 9 i q si i 9 cos g i cos R cos 9 cos q cos i R si i R cos R 9 cos si 9 S i q cos cos R Si R Ide ti fyi g these tw o values f i 8 w e fi d the ter m s m i tt i g the cas e o f i i volvi g cos 9 disa ppear we m y d ivi de by S i 9 d b t ai the resul t si n

9

n r

I

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r s n

S n

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n r

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s

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n

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s n

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which A is ( cos g i

an

S n r

abbrevia t io f i q cos i R) or

n

S n

’

r s n

R ( }

si n

1

R 2)

I

R cos 5 ( R the right posi t ion ’

co s r co s

like m a er w e obtai f 142 i stru m e t cos (X ) cos 8 cos g cos In

n n

n

or

n

n

,

a

S i n r si n

d fo r

the le ft cos (X

a

)

cos 8

Rl

cos R

RI

r co s

an

I

’

si n

cos q

r s n

R 1 si n

co s

q Si n

r Si n

R2

i

R

cos R R i g cos s i R i n R Ide ti fyi g t hese e pres i s w have A cos ( R + R ) 0 B u t w e have alrea dy see t hat co s r

2

s n

x

n

n

r

2

S

e

s on

:

2

1

.

n

A si n

by squari g cos i ( q n

an

s n r

d

} ( R1 + R 2 ) 7

:

0

,

a ddi g w e see t hat A 0 or i q cos r si R ) i 5 ( R R ) cos cos R cos } ( R n

s n

n

’

s n

r

,

’

cos

n

’

I

2

’

1

1

’ .

R2 )

.

Of

t he

RA L I Z E D

T H E G EN E

I N ST

RUME N T

[ CH

XX I

e ter o ly i t he co m bi atio R R the i de error f circle I has bee eli m i ate d We t hus ob t ai a form ula i ter al co s t a t s q n d m y be f d by S howi g how t he t w d 8 are abse t the for m ula d oes bservatio A t d e pe d t he s t ar or m ark chose w hile X d 9 w hich d efi e the as pe t o f t he i stru m e t al o va ish If f brevi t y w e write A s RI

an

d R2

n

n

n

n

n

o

o

n

n

O

S a

.

n

n

n

n

n

n

s

,

a

n

a

r

x

ou n

n o

,

an

,

n

2,

n

n

n

n

I

.

an

on

n

n

n

,

c

.

or

A = si n

} 1

—R R 2) ; ( 1

cos R cos } ( R d be e resse p y ’

'

O

the equatio

n

ma

A

— 4

'

-

I

R 2) ;

R2 )

x

co s

q

si n r

+

B si n q c o s r + C c o s r = 0

,

’

i volve o ly qua ti t ies k o w by observatio T h e a m e o peratio applie d t o a o t her star or m ark w ill give a si m ilar equatio = 0 A cos g i + B i q cos + O whe ce CA (B A A B ) i q A O We thus lear i q d co seque t ly there appear to be t wo f which ei t her will atis f y t he require m d l e t l values f q p p co di t io s XV have ho w ever agree d t ha t 90 g is t he i cli atio d it is a co n ve t io ( p 33 ) t ha t the a gle f circle II t o circle I i cli atio S hall lie be tw ee 0 d so t hat q d e oti g d A ccor di gly w e 90 d is t i guish m us t be be t wee w hich f t he two u pple m e t al a gles m ust be take d thus h m fi is k ow wi t u t a bigui t y We d also q i n w h i ch A B , C

n

n

,

n

n

s

n

n

n

.

n

n

’

’

s n r ’

s n

’

co s r

’

n

an

S n

r

S n

n

n

’

,

’

n

'

or

a

n

e

u

O

s

°

n

n

e

.

,

n

,

an

o

n

n

an

n

n

an

n

o

A O) t a n ’

ca n

n

n

n

,

an

n

.

(A C

an

n

s

’

°

n

n

.

n

°

on e o n

n

n

n

n

n

n

”

(B C ’

r

BC

) ta n

q

.

Fro m this is k o w f as betwee d 1 80 we choose the value which lies as m ust lie bet w ee d 90 T hus q d t h e two i ter al co sta ts f t he ge erali ed i stru m e t m y be d eter mi e d n

r

n

n

n

,

r

n

r

n

n

n

a

.

De

n

°

n

n

o

n

ca n

,

X si n 8 + Y c o s 8 c o s a si n

X

o

n

n

z

.

o

w here

an

.

r

e

r

n

,

,

an

te m i n a ti o n o f X a n d 9 f these qua t ities m T h d eter m i a t io be w ritte f for m ula ( iv ) f 1 4 3 which 147

o

or

r

,

an

,

°

cos 9

,

Y

:

+

Z

q

co s

si n r

si n

a

be m ade by ea s m y n

n

8 si n

a

cos g

9 Si n X

,

Z

c o s r si n :

si n

R 9

’

0

cos X

.

( i)

,

TH E GE NE

R AL I ZE D

I N ST

RUM EN T

[C H

XXI

’

.

t he in stru m e n t be reversed a n d d irec t ed again o n the sam e s t ar at 8 w e k n o w that R is chan ged in to 1 8 0 R the readin g R bec o m es R a n d y is u n altered w he n ce cos r c o s R M s i n ( R y) N cos ( R y ) ( iii) I n equatio n s ( ii ) a n d ( iii ) both M a n d N are k n o w n fo r as t he e observa t io n s give Th lace o f the s t ar is k n o w n a 8 are k n o w n p R R R T here are t here fore t w o lin ear equa tion s i n Si n y a n d cos y i n which t he c o e fficie n t s are k n ow n From t he s e s i n y a n d cos y are d eter m i n e d so that y is a scer t ai n ed w ithou t a mbiguity We have thus S how n h o w all t he co n s t an ts o f t he gen eraliz e d i n s t ru me n t m a y be ascer t ai n e d If

°

’

’

,

,

,

2

,

,

’

2

2

.

,

.

,

’

,

,

2

,

.

.

.

,

.

*

14 9

On

.

a

si n

g

le

e

u a ti o

q

wh i ch

n

th e o r y

m pr i s e s t h e

co

t a l i tr u m e n t s o f t h e O b e r v a t o r y L t 8 be t he co r di a t es f a star S f whi h t he rea di gs f the ge erali ed i t ru m e t are R er let a I like m a seco d s t ar S with coordi ates have t he rea di gs R R Th e oor d i ates m y be altitu d e d a i m uth or right asce sio d d ecli a t i or la t i t u d e d lo gi t u d e or y other sys t e m For t he cosi e f t he a gle bet w ee t he two stars we have the e pressio si 8 i 8 cos 8 cos 8 cos ( 0 ) w hich m y be w ri t te si 8 i 8 cos 8 cos 8 cos { (X ) ( X 142 w e b s ti Fro m the ge eral form ulae f — t ute i the e pressio jus t w rit t e f s i X si ( ) cos d cos (X — ) cos 8 t heir equivale t s i ter ms f R d the co s t a ts f the i s t ru m e t 9 g er w e c I like m a 8 cos cos c their substitu t e fo i n Si X X a ) ) ( ( equivale ts i ter m s o f R d R d t h us obtai d 9 q e pressio f t he cosi e f the a gle be t wee t he two s t ars i R R; term s f R d t he co ta ts f t he i stru m e t Th e work m y be i m plifie d by observi g that 9 ca ot e t er i t o the result f it is bvious tha t t he a g le betwee the t wo s t ars m us t b e i de pe d e t f t he positio f t he fu d am e t al circle w i t h regard t o w hich the coor di a t es are m easured It i t h ere fore per m i sible f this par t icular calculatio t o assig t o 9 y arbi t rary value we please wi t hout restricti g t he ge erali t y o

f t h e fu e

dam e n

n

a,,

O

,

a,

on

an

o

n n

’

n

,

z

n

2

,

.

n

n

,

an

,

n

2

.

n

n

n

,

a

S n

2

,

s n

,

2

,

a,

n

n

n

s

,

n

an

r

an

,

,

an n

n s

n

O

n

n

o

n

.

n

nn

n

O

n

n

n

s

2

n

n

n

os

n

n

n

an

r, an

,

n

O

n

an

an

nn

n

.

S

or

,

,

,

a?

2

an

a

,

’

o

,,

,

O

su

a,

n

c

2

or

n

n

n

n

n

n

or

n

n

O

ca n

,

n

,

r

o

,

o

x

n

x

a

a,

2

n

n

a,

2

n

n

an

n

c

n

an

,

.

or

,

,,

n

n

n

,

a

n

x

o

n

,

n

an

n

n s

z

n

s

o

,

n

c

n s

s

.

or

n

n

n

,

n

R

148 — 1 4 9] o

the result llo w s

f

fo

Z

TH E G E N E A L I E D

we

If

.

m

ake

9

I N ST

90

RUM E NT

the equa t io

°

be

n

co

cos 8 cos 8 cos ( ) i 0 i R cos R R i q B g x cos g i cos R cos i R cos R R i q i R cos cos B R i g cos i R ( sg i i x ( cos i R c cos R cos R si g cos i R q i g i c os q cos i cos cos si i g q which gives the foll wi g fu da m e t al equa t i si 8 si 8 cos 8 cos 8 cos ( ) si n

8, si n 8,

co s

,

s n r 0

8

s n r s n

s n

s n r

si n

n

,

2

q

"

co s

co s

2

2

co s

sin

Si n

2

g Si n

2

co s r co s

,

si n

s n

co s r co s

,

si n

’

,

S n

r s n

,

si n

n

r s n

,

si n

’

,

,

n

n

,

s n

,

co s

,

n

S n r

n

,

i

r s n

on

a,

,

a,

r

cos (B

r

R,)

,

R sin R g r co s R cos R cos ( R R ) c o s r Si n R Si n R c o s (B q r sin R Si n ( R si n R ) q ( ’

c o s r si n 2

’

,

,

’

’

,

”

,

r S n

o

n

co s

,

os r

,

_

’

’

r

,

S n r S n

,

r S n

,

es as

a,

r S n

,

S n r

co

a1

,

m

,

,

’

2

,

’

,

,

R,)

,

’

c OS

,

cos g

Si n r

cos

r Si n

,

,

R,)

(R1

(

co s

R,

’

cos B 1 i R R i ) ( g { }( f ircle I i i t pla e m us t be I t is bviou t ha t a r t ati wi t hout e ffe t t he dis t a ce S S H e e R d R e ter i to d o eque t ly t he R S S o ly by t heir d i ffere e R i de error f cir le I does t e t er i t o t he e pre i We m ight i d ee d have further abbrevi at e d t he work by m aki g R 0 be fore m ul t iplyi g to f r m t he eq a t io ( 1) i f a ft er t he m ul t i pli ca t io w e re place d R by ( R We m y u ppose tha t t he i de err r f circle II is A i w h i h case R d R h oul d be re pla e d by A d R A We have alr a dy sh w how A d le ft bserva t i s f t he a m e bj e t m igh t be f u d by right It m y however be d e t er m i e d otherwise as will prese t ly appear d this for m ul be B y assig i g sui t able values to q m a d e to a pply to the follo w i g astro o m ical i s t ru m e ts —the alta im u t h t he m erid ia circle the pri m e Ver tical i s t ru m e t the equa t orial d the al m uca tar W e shall see la t er t hat f the shoul d be each as ear ero as possible m eri dia circle g d d qui t e arbi t rary d fo t he al m uca tar q is the lati t u d e Th followi g ge eral proo f will Sho w t hat the co m plete t heory f for m ula each f the i s t ru men ts a m e d must be i cluded i this si n

c

,

,

o

s

O

co s

i n r cos r s q

co s

on

n

on

,

c

,

n

,

n

an

,

an

,

s n

,

s

n c

.

n c

n

,

s n

,

o

’

c

n

,

n s

n

n

~

x

n

n

n o

c

o

n

x

ss o n

n

n

o

n

n

u

s

a

,

,

x

o

o

n

,

,

n

’

n

.

c

’

an

,

,

S

’

an

c

o

a

,

n

,

,

an

n

n

,

,

an

an

an

r

e

o

n

n

s

O

c

n

r

a

,

n

.

.

,

n

ca n

n

n

n

,

or

.

z

n

r

an

n

n

o

,

n

n

o

,

n n

z

on

O

an

n

e

.

,

r

.

n

o

n

n

n

on e

2 9— 2

.

TH E

G E NE

RAL I Z ED

RUM ENT

I NST

[OH

XXI

.

Fro m y such i s t ru m e t w e d em a d n m ore t ha t hat the two read i gs R a d R O b t ai e d by directi g t he i stru m e t t ar t icular star shall e able us to calcula t e t he coordi a t es 8 y p f t ha t star free fr m i stru m e t al errors L t S S S be three sta d ard stars f w hich t he coor d i ate are k ow d le t each f t h e e s t a s be bserve d wi t h t h ge erali ed i s t ru m e t wit h re ults R R R R respectively S bs t i t uti g f each o f the three pairs ( S i t h e ty pical for m ula ( i ) we ob t ai three i d e (S m d e e t equa t io s Fro these equa t io s be d A n p g will t here be y i d fi i t e e i the solutio fo i N fou d each c se w e m y regar d t hese qua ti t ies as a ppro i m ately k ow so tha t t o obtai t he accurate values f g d A we hall have t o solve o ly li ear equa t io s We m y t hus regar d ( i ) as equatio co ecti g R R d k ow qua ti t ies We L t S be t h e star whose coor d i a t es 8 are sought d subs t itu t e their w rite the equa t io ( 1) f t he pair ( S R We t hus have equatio n u m erical values f co ec t i g t he coordi ates 8 f y star w i t h i t s c rresp di g R R d k ow u m erical qua t itie Whe we subs t itute f R d R t h values bserved f S the for m ula re duces t o a u m erical rela t io be t wee the d 8 f the particular tar S Fro m t he pair ( S S ) w e fi d i like m a er a t her qui t e i depe de t u m erical equatio i volvi g 8 A ho w ever t w equa t io s are o t ge erally su ffi cie t t deter m i e 8 without T his equa a m bigui t y w e b t ai a t hird equatio fro m ( S ti is t i de pe de t f the others bu t i f we m ake = si 8 h h cos cos cos 8 w e all obtai t ree li ear equa 8 si y f d 8 f d t io s i by t he solutio w hich are ou w ithou t a y a m bigui ty ectio wi t h t he di ff ere t A l l the or d i ary for m ulae used i co i n stru m e ts a m e d be d e d uce d as particular cases o f t he ge eral equatio mal l qfi t i t i uch th at th e r ec d d S h w th at i f g d b E f ll w higher p wer m y b m itted f r mula ( i ) m y b w ritt n

n

an

’

n

n

n

o

n

n

n

n

n

n

an

o

o

e

,

n

n

,

,

z

n

n

n

o

s

o

n

u

n

n

n

’

,

n

n

e

r

,

n

a,

an

,

ca

n

,

r

,

x

O

n

n

n

n

nn

n

,

ss

n

n

n

n

n

n

,

,

,

a

n

,

or

an

a

,,

,

.

or

.

e

,

n

n

s

O

n

,

n

r

s

n

.

,

.

n

an

,

a

n

3

,

o

n

r an

n

,

S

a

.

n

an

a,

,

an

,

n

n

.

n

or

or

,

,

n

an

n

n

,

n

n

on

n o

a

n

n

a

,

,

o

s

n n

n

n

.

n o

a,

n

s,

.

o

n

o

,

n

a,

n

n

n

o

S

n

z

n

a

,

n a,

z

,

an

n

n

or

,

n

O

,

n

on

n

.

n

n

n

s

a

n

o

or

O

n

n

an

n

e

n

n

o

n

n

’

an

,

a,

n

an

.

an

a,,

n

n n

’

a,

n

e

n

n

n

an

o

,

.

n

n

n

n

n n

n

n

can

n

n

x

an

o

.

an

e s

r

1

es s

on

s

‘

o

si n

s

a

e o

8, S i n 82 + c o s 8, co s

R,

+ g si n

’

,

co s

8,

o

co s

(

a

,

—a

(R l R zH — R si n R R ,) ( , ( , cos

R,

’

co s

—

'

-

2

e n as

e

a

o

o

s :

)

si n

R,

’

si n

R2

’

’ — — co s c o s R R R Z) ( , ( 1

an

T H E G ENE

be t he ositi p

If N t he

n e

’

w

o

bu t

O

n

RUM E NT

n

n

n

n

N N N = 18 0

’

N N

NN

’

°

”

n

n

n

7

s

NN

N i

7; s n

,

T

hus we obtai

AX

i

the

n

cos 9AR cos 9A R

,

.

three for mulae A9 = n

S n

,

N N

8

t he

n

MN

”

AX

; si n

.

n

NS s i n SN L 7

n

,

’”

”

d N

co ec 9

a

’

XX I

-

s n

n

,

SN

i

an

.

,

N NN

7; S n

,

CH

9

’

si n

on

cos S N L whe ce Si 9A R cos 8 ; i (X ) I f fi ally a perpe d icular N N is draw to 9A R

Si n

or

,

[

n

o

”

’

A "

I NST

f ositio t he poi t origi ally a t N p f t he asce di g o d e f N V M N the w

n e

on

RAL IZ E D

9A R

co s

i

7) s n

— X (

(X

) cos 8 01) cos 8

a

cos 9A R ) i 8 W e shall w i ves t igate t he e ffect u p n d 8 f a cha ge f 9 i to 9 + A 9 i t bei g su ppo e d t ha t X g R R re m ai u altere d w hile t his cha ge t akes place T h cha ge is f course equivale t t o a r ta t io f t he fi g ure N VK S rou d N through a n a gle A 9 w hile this figure re m ai s u altered i for m N S is u cha ge d d S m oves per pe dicular t o N S through t he s m all d is t a ce si N S A 9 it is obvious fro m the figure tha t this i crea e o f 9 di mi ishes t he d ecli atio by AX

1

n o

o

n

n

o

s n

n

,

n

s

n

o

n

.

O

O

.

,

n

N S s i n N SL A 9

We thus bt i

n

,

,

n

n

n

Si n

r

n

n

s

’

n

an

n

n

n

o

n

n

o

e

.

n

n

,

,

n

n

a an

si n

n

(X

at

) A9

.

a n

We have als

A 8 = —si u

— a A9 X ( )

o

cos (X

a

) ta n 8 A 9

l t o f t h e di ff e r e n t i a l f o r m u l a e T h for m ulae ( i ) ( ii ) d ( iii ) 1 5 0 will e able us o w to d e d uce fro m the fir t f t he for m ulae f the ge erali e d i s t ru m e t 1 4 2 t he re m ai i g for m ulae ( 2 ) d 5 Th first for m ula is *

15 1

A pp i c a i o n

.

e

,

,

s

an

,

o

n

,

.

n

,

n

or

n

z

an

e

si n

8= —

co s Si n co s

Si n si n

9 si n q si n 9 i n s q

r

cos cos R 9 cos g cos i R 9 cos i R cos R 9 si g cos cos R s i R r

r s n

n

’

’

r S n

r

n

n

’

n

n

1 5 0—1 5 2]

t his

T HE

G EN E

RA L I Z ED

UM ENT

I N ST R

45 5

ust be u iversally t r ue i t m ust be t rue i f 9 be i creased by A 9 while 8 receives its c rrespo di g variatio Perf r mi g the di ffere tiatio substitu t i g f A 8 from ( ii ) d dividi g by A 9 w e have i ( X — a ) cos 8 = — i 9 i g s i cos 9 cos q i cos R si 9 cos q cos s i R cos 9 cos S R cos R cos 9 i g cos cos R i R T hus we see ho w t he first leads t o the seco d f t he fu da m e tal for m ulae 14 2 Fi ally le t the equa t io j ust ob t ai ed be sub mi t te d t o the d i ff ere t iatio as alrea dy e plai ed i n 1 5 0 wi t h respect to A 9 A X A R all o t her qua t i t ies re m ai i g co sta t d w e have cos ( X ) 8 AX i 8A9 i S i R cos cos R B ( q i g cos S i R i R ) cos 9 A R E li m i ati g A 9 A R A X by equation ( i ) 15 0 w b t ai t he third f the t h ree fu da me tal for m ulae f the ge erali ed i stru m e t 14 2 i cos ( X ) cos 8 cos g si r s i R cos cos B cos R As

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e pressio s f t h e right a ce si d the d ecli a t io 8 f the n ole f circle II i the ge erali e d t ra sit circle we observe that i f had bee 90 the tele cop e w ould be n eces arily always poi t e d t o 8 a d there f re by substi t utio n f 90 f i equati s ( i ) they m us t be sa t isfie d by a 8 w he ce — i i n cos 8 s (S + X + q 8 cos 9 cos 8 si X ) 1 gs From the fi rst w e obtai 8 q fo w e rej ect t he solu tio 18 0 9 becau e T h secon d equation requires 0 3l q :l q d f t hese t h for m er that S + X shall be ei t her 90 is i ad missible f it would o t sa t i fy the third equatio H e ce fo t he coordi ates o f the ole f circle 1 1 we have To fi n d

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to t he la t ter there is cer t ai m ethod f comple t ely eli mi a t i g their e ff ect u less by t he act al d e t er m i a t io f t he error o f each se parate t rait all rou d t he cir u m fere ce followe d by a rigorous applica t io f its err r t o t he readi g f every trait i volve d A s t his w ould require a se para t e i ves t iga t i f ea h o e o f several thousa d traits the task w oul d be a colo sal e t ge erally a t te m pte d T h e errors f i divid ual d i d ee d is t rait are t este d a t di ffere t par t s f the cir le d i f t hey are fou d t be m all i t m y the be hope d t h a t i t he m ea f several observa t io s t ake wi t h everal microscopes t he i flue ce t a ppreciably a ff ec t the fi al re ul t o f t he casual errors w ill A regar d s t he syste m atic errors i t he d ivisio f t he circle t he a sura ce o f t heir d isappeara ce fro m t he fi al re ul t has a m ore sa t is fact ry fou d atio E rrors f t his class m y arise fro m t h e m echa is m use d i t he d ividi g e gi es by w hich t he trai t s t he circle Th t oothe d wheels i t he d ivid i g e e g rave d e gi e are t d ca ot be abs lu t ely t ruly sha pe d d absolutely ce t re d S u h errors i t he trai t s m y t o a large e t e t be d ee m e d perio d ic so t ha t whe t he wheels f the e gi e have per for m e d a cer tai u m ber f revolutio s d a cer t ai ad va ce has bee t he e gravi g t he sa me errors will be re pea t ed T his is m a de i t leas t f t he chie f sources fro m which sys t e m a t ic errors arise i t he places f the t raits d let R A R w h ere L t R be t he read i g f a certai trait A R is a s m all qua t i t y be t he t rue rea d i g f t hat poi t t he circle a t which t he t rait is ac t ually itua t e d T he A R is t he error f t ha t t rait We shall assu m e that A R be repre e t e d by a e pressio f the for m As

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t he i n s t ru m e n t k n ow n as the m eridia n circle or t ra n si t circle by w hich z e n it h d is t a n ces as well as t r a n sits c a n be observe d Th e i mpor t an ce o f t h e m eri d ian circle is however so grea t bein g as i t is t he fu n dam e n tal i n s t ru m e n t o f t he as t ro n o m ical ob s erva t o ry t hat it is use ful t o d evelo p i t s t heory i n a n o t her a n d m ore direct .

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his is the fu d am e t al for m ula f the reduc t io f m eridia observa t io s T h qua tity t is t he rrec t io to be a dde d to the bserve d sid ereal t i m e f the t ra si t t ob t ai t he true i dereal t i m e T his e pressio f t is ge erally k ow as M ayer s form ula I t m y be t ra s for m e d i various w ays For exam ple B essel d eter m i e d by the i t ro duce d t wo w qua t ities m d equa t i T

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t e r m i n a t i o n o f t h e e r r o r o f c o ll i m a t i o n Th qua tity 0 k ow t he error f co l li m ti i t he m eri d ia circle or i d ee d i f f r m o tra sit i stru e t m ca y be d eter mi ed by the i d f what are k ow as colli mati g telescopes f which we shall here d escribe t he use I the focus f the t elesco pe f the m eridia circle is a fram e carryi g a li e w hich be m ove d fro m coi ci de ce w ith t he fi ed m eridio al li e i t o y parallel positio i the la e er e d icular t o t he o t i cal a is f the t elesco e his o T p p p p p m ove m e t is e ff ec t ed by a m i ro m e t er screw wi t h a graduate d head so that by c u ti g t he revolutio s d the frac t io al parts o f a revolutio the dista ce through which t he m vable wire has bee dis placed fro m the fi e d w ire beco m es e actly k ow We sh all first ho w how by t his c triva ce we c ul d d eter m i e the error f colli m atio i f w e could observe t w o dia metrically o pposi t oi ts t h e heave s p 15 7

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F U N D A M E N TA L

1 5 6—1 5 7]

I N ST

R U M ENTS

471

be t he hour a gle d d eclinatio o f a poi t t he celestial sphere the fr m ( i ) we have 8 i l cos 8 8 ( 9 si c = cos l i t) A s 0 l 9 are fi e d qua ti t ies c ec t e d with t he i s t ru m e t it is plai t ha t this equa t i wil l o t i ge eral be sa t isfie d f a give pair f values t 8 T his m ea f ourse m re tha n t he obvi u fact tha t as t he m eri d ia cir le has o ly d egree o f o t be free d m i e r tatio ab u t a si gle a is its teles pe ca directe d to y poi t o t he celestial sphere e ce pt th se w h ich If h wever we give the i stru m e t lie a certai circle a seco d degree f freedo m t he it withi certai li m i t s which f our prese t pur pose are quite arrow li mits be direc t e d u po a y poi t i t he vici ity f the circu m fere ce f C b y the m ovable wire T hi seco d d egree f free d o m i give j us t de ribe d B y m ovi g t his wire t o a d is t a ce fro m t h fi ed w ire d regardi g the i t er ec t io o f t h wire i i t s e w l f o itio wi t h t he h ri ta w ire t he li e colli m atio t he f p t ele co pe t he error f c lli m a t io is w c l w d the equa t io ( i ) be o m es t here f re cos l si 8 si l cos 8 cos ( 9 t) i ( m) is deter mi e d by si m ply screwi g the m ovable T h e qua t i t y t il the a i f t he t elesco pe be direc t ed to t he poi t P w ire o f w hich the co r d i a t es are t 8 u ppose t he t ele c pe direc t ed t o t he celestial w L e t us — f m ro the 1 8 0 which is 8 d t i t w i t h co r i a t es P + ( p t a t the dista ce A gai let t he m ovable wire be for m er poi t P so tha t P shall lie t he i tersectio f the m ovable wire d t he fi ed hori o ta l wire d we have 9 t 8 l i cos os cos l i 8 i (c ) ( ) 0 m) i (c m) i (c n m m as all the qua t itie are s all t hi be wri tt e d y I f t, 8

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F UN D A M ENTA L

4 72

IN S T

RUME NTS

[OH

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x xn

the theory f astro o m ical i s t ru m e t s is e plai ed i the a e ed f the A B is t he t ra si t i s t ru m e t or telesco pe d iagra m m eri d ia circle f which t he ce t ral cube is pier e d by a c yl i d i l hole L M f which t he a is is XX w he t he tele ope is i t h e ver tical positio AB T h a x i about which t he tra sit i stru me t itsel f rotates is perpe dicular to t he pla e f t h e m i a er d the ivot a t the w es t erly e t re i t y is show t he p p p figure while e f the positio s w h ich t h i s t ru m e t m y ssu m e duri g i t s rotatio i i di ated by t he d ot t ed li es Th t wo collim ati g i s t ru m e t s X Y a d X Y are fi ed hori o t ally d cros wires are placed d south o f the m e i dia circle n orth f each f t hese subsi diary i stru m e ts as d F t the foci F i t he focus o f the great i n s tru m e t i t sel f n

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F UN D A M E N TA L

I NST

RUM E N TS

[

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x xII

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agai through t he obje t glass i t he opposite direc t io d will there fore f r m i m age f t he cro s wires at t h focus be si de t he wires t hem selves We have the o ly t o shift the m vable wire t hrough such a m easure d d is t a ce 0 as shall m ake t he i tersec t io f t he cro s w ire coi ci d e wi t h its reflecte d i mage d the we k ow t h a t t he a is o f t he t elescope m us t be d f er e icular to t he sur f ace m ercury m us t t here ore p i t f d p p t o t he d ir Th e d ecli atio f t he a dir is l w h ile i t s hour a gle is Wi t h t hese substi t u t io t he equa t io ( i ) re duces to i (c c) i b d there fore b = c + c as all t he qua ti t ies are s m all the f sol t i 1 8 0 f m us t be rej ecte d T hus b is k ow c t he error o f colli m a t io is su pp se d to have bee previously fou d d c is as alrea dy s t a t e d the qua ti t y which has just bee m easure d mi tt e d

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FUN D A ME N T AL

1 5 8 — 1 60]

1N ST R U M EN TS

f ossible It is there ore ecessary tha t f the t w decli a t io s p shall be ear ero d t he o th er n ear H e ce we lear t he i mporta t prac t ical rule t ha t f d e t er m i i g t h err r f t he cl ck d the a i m uth f t he i s t ru m e t f t he stars ho e shoul d be n ear t he p le d the o t her sh ul d be ear t he equat or I t will be observe d tha t while b d 0 be f u d wi t hou t observa t io f celes t ial bodies this is o t t rue with regard to A T n

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I t is t he fu ctio f the m eridia circle t o e able t he observer t o d eter m i e bo t h t he righ t asce sio d t he d ecli a t io f a celestial obj ect a t t he a m e tra si t We have already how how t he righ t sce sio is f u d d i t re m ai s t ho w how the d ecli a t i is m easure d A early as pos ible a t t he m o m e t f ul m i a t io t he h server m oves the telescope tha t t he star appear t o alo g the hori o t al wire s t retche d acros the focu f t he telesco pe Th circle is the n t be read by t he m icr sc pes i the m a er already e plai ed I t is esse tial tha t a t lea t t wo m icro sco pe t opposi t e e ds f a dia m e t er be e m pl ye d but f ur micr sc pes sy mm etrically placed ro d t he circu m fe re ce are require d f t he bes t i stru m e t d so m e t i m es eve m re tha n T h m ea R f t he rea d i gs f t h e e m icroscopes four are use d is the ad opt ed as t h e rea di g f this par t icular ob erva t io ( see n

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FU N D A M E NTA L

476

INST

RUM E NTS

[

OH

x x II

.

d thus t he true e ith z d is t a ce is re fractio ( C hap A ssu m i g t hat t he lati t ude qt is k o w w e h ave the fou d — fr m t he equatio 8 d ecli a t i j It is o ft e possible to dis pe se w i t h observatio o f the f s t ars who e d ecli a t io s are alrea d y n adir by m aki g use k ow I f uch a tar be observed wi t h the readi g R wje ob t ain R as t he e pressi o f i t s a ppare t e i t h d is t a ce R d agai correc t i g f re frac t io we ob t ai the t rue e ith d is t a ce B t t his is j> where 8 is t he s t ar s decli a t io n} a d thus i f d e o t e the re fractio se e

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erved ti me f tra it f a k w tar wh e decl i ati i 30 i f u d t b c rrect i t agree with t h tar right a ce i whi le th b erved ti m f tar i decl i ati 15 d 60 f u d t b 7 4 d + 31 5 r p cti vel y i err r Pr ve that t h err r t b expected f a tar i decl i ati 4 5 i ab ut 1 1 [ Math T r i p I ] el f rm ula ( i v) 15 6 w b tai t h f ll wi g f u r equ ti U i g B d t h re u l ti g equ ati fr m which m b el i m i at ed f X w i ll g i e t h d i red re u l t m + ta 30 + 30 Ex 1 .

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a tra it i tru m t f 10 fe t f cal l gth which i c rrect except f c ll im ati err r a tar f decl i ati 60 i b erv d t cr t h mer i di a 2 t wire Sh w th at t adj u t t h i t r u m e t t h c r m t b m ved a d i ta ce 0 0 0 8 7 I w hich d i r cti h u l d t h wi r b m ved ? Ma th T r i I p [ f c ll im ati i wh e ce T h c rrecti Th circul ar m a u re f thi a gl 10 feet g ive 0 0 0 8 7 If w rememb r that a tr m ical t le c pe i rever d right d l ft it i b vi t h i m age i th at t h w i re mu t b m ved t t h ea t Ex 2 .

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h w th at a t ra it i t rume t c uld c ce i vab l y b th at all tar pa i g t o t h uth f t h e e ith w ul d appear t cr i g t h meri di a b y a c ta t err r 1 Ex 3 .

.

S

s

o ss n

s

e

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n s

ss n

e

n

n

n s

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adj u ted b late i ll E xam ] e

on o

c

[ Co

.

so

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s

e

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.

F U N D A M E NTA L

wri te t h eq uati

W e may

e

t+

co s

i er t val ue

d t2 b e t h e t wo d ff

an

whe ce b y ub tracti n

s

s o

en

C= O

.

which ati fy thi

ft

s

co s

A s i n t2 + B

c o s 152

+ O = 0,

t by

si n

ivid i

g

ou

5

( t,

n

tan

x xI I

.

’

A s i n t1 + B

d d

an

on

[O H

15 6) i n t h e fo m

A sin t + B

If t,

RUM ENT S r

( ii )

on

IN ST

s

uati

s eq

on

t1 +

5 (t1

h ave

we

t2 ) = A /R ,

while t h c ll i mati e ter i t C l y E f k w latitude j 9 A t ra it i t ru me t i m u t d a t a plac i a ver tical pla which i t t h m rid ia Fi d equati t det rm i e t h a i m uth A f t h i t r u m e t i term f t h b erved ti m 9 b twee tw ucce ive tra it f a ci rcum p lar tar f decl i ati 8 S h w th at a m all err r A 9 i t h b erved d i ffer c e 9 f h ur a gle wi ll lead t err r i t h a i m uth f mag itu de o

e

x

.

on

n

ns

.

n

s

z

o

e

s

o an

o

e

5 si n

2

A

A

ta n

n o

o

an

o

e

b

e eral r u la

ec e

on

en

which m

a

wr itte thu

y be

n

T

hi

m

s

= Q we 9 , 5

u t b t r ue i f h ave t h t w e

(P +

1 t an A

co s

(P

u tracti

S b

M

ul tipl yi

g ( i ) fr

n

n

o

( i ) by

g

o

n

whe ce fr n

n

theref r R st r i g t

d

.

n

o

e

o

Q

s

Th e

regard

se

r ula

d t h e fo m

co s

t = o,

s

co s

ub titut s

ed

qt t a n A t a n 8 fo r t

an

d m

.

aki

n

— t g

d d A

n

§

9= P

an

d

e

ivi di

n

g by

Q

si n

ob

we

P + co s P = O

si n

A tan 8

ta i

n

0

.

Q ) a n d s u b t racti n g ( ii ) m ul tipl i ed b y s i n (P 2 Q = 2 c o s gb t a n A t a n 8 c o s P s i n Q

Q)

,

,

Q

co t

I +co t

val ue 59 1

2

( ) si n

1

,

2

1

1 tan

A= 2

A)

we

— co s ( ) co s e

o

on

8s

co s o

2

i

is

o

an

b

P,

1 ta n

2

8

.

n

5 9/(c o s 5 9 A

n

()

2

en

s

Q

s

e

e

2

a i l y b tai

2 = A cos

on

8 cos P,

= c o s ( ) ta n Q

tan

2

d 9

.

tai ed n

i r tiati

b y d ffe

en

n

g A

with

.

.

n

.

to 9

E x 10

ce tre

si n

e

on

t=

si n

t

r qui red equati b twe c d part f t h que ti

th e

l

8

co s

Q)

si n 2 ( )

co s

which i

A

c) S i n

= Q ) c o s (1) t a n A t a n 8

1

i ts

8 S i n t + s 1n

an

(P

Sin ( ) co s

(

e

o

0 , b = 0 , [z = A

c=

sin

(P

co s an

ake

ri

Q

g b y si n

(iii )

om

s

T p

.

Q)

pta

A

n

.

Q ) = c o s (1) t a n

si n (

1tan

o

(P +

an

si n

.

sin

ii

m ( )

S i n