151' 



PROBLEMS IN SPHEROIDAL TRTANGLES. 



Now by fphcncs f — — - j is = cof. &'* fere, and (in &' == 

 fin a— 2 fin «'^ cof. «■' col. >.. — ^ — .S, by theorem, page 17 ; or 



becaufc ■ = col. ^ cotang. a (er6, the fm a' is = fin o — 



2 fin «•' cof. a'* cof ?^^ tJ ; confequently w is = 

 /Zfincif fm A^ 7 ( , 2 cof A^ cof a* t 



From the theorem fin a,'-— fin «• — 2 fin a' cof a'^ cof. x*. o, 



it is manifed that a is nearly equal to a'-}- fin 2a' cof X-. S. 



When « is nearly equal to a right angle, we have w = 



tZ fi n <K f . ^ , "1 D fi n a , . , 



■ — : — < l — o fm X^ (■ = — ■ , which correlpouds with one 



coi.(P I J Kcul.c? ^ 



of Legendre's theorems. 



prob. IV. 4. • Having given the laticudes of two places, and their 



Given two lats. jifiei-ence of longitude, it is required to find the horizontal 



3nd dift. long. ° '■ 



To find horiz. angles ? 



ang'es. L^t X, (?> be the latitudes of two places, u their difference of 



longitude, and «, [3 the horizontal .angles at X, <p refpeflively 



on the fpheroid ; alfo a, /3 the correfponding angles on the 



fphere, which may be found from the data by the rules of fphe- 



rical trigonometry. Th^^n by the theorem, page 17, if we put, 



r /2 ^"^- ^ fin (f- fin X 



M = 2 fin c&'2 X — F— X ^7- = 2 fin a'- x 



: col. (p lin w 



cof X cof 



2 $ — X 

 — -— X fer^-, 



CO!. ?> w 



N = IVi. ^ fin X (fin X-{-fin <?)— ~ > + cotang. a' M*, and 



N' —. AI. j - — fm <P (fin x + fin f) ? + cotang. /3' M^, 



we diall have «=a'-}- IVU -{- N ^, and /3 z= ^'-M 3 + N'^. 

 iLxnmpIe. Let a ::= 3496740, bz=.3477210, X — 49''40', 

 <P ZZ 50° 0', .and !^ ZZ 30'. (See the Account of the Trigonome- 

 trical Surveij, <.V<^"- Vol. 1, Page 158J. Then the two colali- 

 ludes, and the included angle .30', will give the fpherical angles 

 «', (2', 43° 51' 48^.3, and 1.35" 45' 16",2 refpeaively. The 

 jcniaiiiing part of the calculation is as follows: 



2 fin 



