PROBLEMS IN SPHEROIDAL TRIANGLES. 157 



M -f- ^, gives Szz , very nearly. Her6 a queftion 



naturally arifes ; viz. To determine the value of »' fuch that 

 3 may be obtained in this manner with the highefl degree of 

 accuracy, of which the method is fufceptible. 



This will evidently be the cafe when M is about Its maximum 

 value. So that if we put the differential of M IZ o, and fubfti- 

 tute the value of the differential of a^ refulting from the proper- 

 ties of the fpherical triangle in this equation, we fhall have the 

 required value of the horizontal angle. Let us fuppofe X ant! 



, , ■„ /- , ^ ., fi" ^ — fi" ^ 



^ given, and there w:II relult fin a'* x ;: a maxi- 



cof. <P 



mum; whence 2 cotang. a'cof. <P (fin <p — fin x) c?a'4- (I — 



fin ^ fw <P} X d<Pz:zo. Butby Ipherics, 



„ , , cof. X tang. <p — fin A coC. oj 



Cotan?. a,' — — . ^1— — , 



^ fin*, 



J J , fin «.'* cof. X , , - , r t n- ' 



and oa'zz ;; — X " ?> ; wherefore by lubftitution, 



cof. ?>* fin w ^ 



^- , - , cof. I? (1 — fin X fin (p) fine.. 



2 fin ct' cof. a'zz ■ :: ■ X 



cof. A fin ?> — fin A. 



Alfo when « and <? — X are fmall, the fin v zz --^ — -r • 



col. 9 



I fin (<? — X) -\- 2 fin A X cof. (p fin -^ w' ?■ , and fin ?>— (in ,% 



■2 4- X X 



•:ZZ2 cof. - M X fin — ^ — ; confeqaently 2 cof. «'*;;::;: 



{I -^ fin ^ fin a) fin (^ — X) -f 2 fin X cof. <? fin | a;' 



2 cof, X cof. (^j*" 2(in(^i^') 



and 2 cof. a'^ ::;^ j if vve fuppofe .<? zi x nearly, and fin | a>? 

 very fmall. 



When the horizontal angle .»', therefore, is equal to 45° or 

 133**, the obfervations will be nearly in their moft favourable 

 flate for determining the comprefTionby means of our theorem. 

 We might illuGrate this method by examples taken from the 

 trigonometrical furvey of Great Britain., but on reference to 

 it I have found fewer complete fets of obfervations than might 

 be expeded, and fuch as are complete in every refpe6i, are 

 not well calculated f<?r this purpofe, the horizontal angles be- 

 ing 



