156 mOBLEMS IV SPHEROIDAL TIIIAKC)LE$. 



<P —X 



ably fimplified. For then M~2 fin «'* cof. x x — -;-, and 



N 3: M (2 fin X* — i) -f- cotang. a' M' ; and if a = 90° fer^, 



(» — X 

 M is — 2 cof. X -j , and Ntz: M (2 fin x' — |). 



The particular flates of the data, when the term involving 

 the fecond power of i is rigoroufly equal to nothing, may be 

 thus determined : 



In the firfl place, if ozzo, or the eccentricity of the fphe- 

 roid be infinitely fmall, », «', and ^, 0\ are exa6liy equal to 

 each other. 



Secondly, when M ZZ o, N is zn 0, but M is zz when 

 !p rr X, or the fine 01,'zzo, that is when the triangle is ifofceles, 

 or the directions of the places due north or fouth of each other. 

 It is almoft rl- It appears then that this property of fpheroidal triangles, firft 

 goroutly exaft. ^(jvgnced by Mr. Dalby, andobjeded to by Mr. Play fair, isalmoft 

 rigoroufly exad ; anil it might eafily be (hewn, that its appli- 

 cation will never occafion any material error, even in the moft 

 ^nfavourfible cafe that can be propofed. And it is not merely 

 an elegant and curious theorem, but is highly valuable, as af- 

 fording a method of determining the longitudes of places from 

 terrefirial meafurements, almofl: independent of all hypothefis. 

 For whether the earth be an exa61 ellipfoid or not, any fmall 

 portion of its furface may certainly, without error, be con- 

 lidered as pertaining to one of fmall eccentricity, which 

 fuppofition is all that is neceflary for demonflrating the 

 theorem. 

 Dctcrniinatlon Our folution alfo afFords an eafy method of determining the 

 or the ccceiitn- eccentricity at the place of obfervation. For if we have the 

 obf. latitudes and difference of longitude given, we iliall alfo have 



the horizontal angles on the fphere. But from obfervations of 

 the pole ftar, we may find the horizontal angles on the fpheroid, 

 and confequently the di.Oerence between them ; but this differ- 

 ence is equal to a certain function of ^ in our folution, whence 

 we fliall have an equation, from which 3 may be determined. 

 Thus if a be the obferved horizontal angle on the fpheroid, and 

 a' the computed one on the fphere, we have M^ -f" ^ ^* ^^ 

 «• — «,'. Now if we reject the term N ^^ as infenfible, we ob- 



* — «-' . 

 ^ain a near value of — — r-; — , which fubHitutcd for m 



