PROBLEMS IN SPHEROIDAL TRIANGLES. 15^ 



! Q fin a" -Log, 9.9S242- Sin X (fin ^-|-fin(P)-i 0.665036 

 ' Cof. ^ - - Log. 9,81 106 - - Cotang. a'.M - - - 0.432644 



Cof. -P- Log. 9.80937 - - Cotang. /3'. M - - - 0.426905 



Sec. ^ -- Log. 0.19193 



(p-X - - - Log. 1.30103 N - I.og. 9.65937-N'Log.9.65937 



w cour. - Log. 8.52288 206264",8 - Log. 5.31443 



^ - Log. 5.49408 



M - - - - Log. 9.61889 



206264^8 Log. 5.31443 M"^* - Log. 0.467 88 -2^^937 



^ .Log. 7.74704 



M"^ - - - Log. 2.68036 - 479'',025 

 +2 ,937 



481^',962 — 8' 1^',962: 



Therefore air43<' 59' 50",2G2, and /3zz 135^ 37' I4'',238. 

 Mr. Dalby makes a— 43 59 51 , 55, and /3zz 135 37 12 , 95. 



I have already remarked, that the fum of the horizontal To afcertain the 

 angles on the fphere and fpheroid are very nearly equal, and ^^° ^^ error of 

 that they would be perfectly fo, if we were permitted torejeft that the fum of 

 the terms of the formula involving the powers of ^ higher than t^«^^oi"'z- angles 

 the firft. We fiiall now, by retaining the fquare of ^, afcertain and fphemid are 

 the probable error of (his theorem. nearly equal. 



We have then by the formulae, « + /3=a'+/5'+ (N -f N')^». 



Now N+N'is equal to M ( fin x^-fin $2+5^^.--^-- 1 

 I fin «.' fin jS' J 



(■ 2 fin (x'-l-^') 7 



— M (fin X — fin (p). ^ fin X — fin ?> p— ?— [ ; but in 



L fin w J 



all the cafes that occur in pra61ice, A and (p are nearly equaf, 



and the fum of «.', /3' differs little from two right angles, where- 



fore the fin a* — fin £p*, and — : y—. — ^r;-- muft be fmall, and 



tin a. lin p' 



the fum of thefe is not only to be multiplied by M, which is 



alfo fmall, but by ^*, which is about . So that (N + 



^ 90000 ^ ^ 



^') 3' is infenfible, and therefore a + /3 ~ »' -}- iS'. 



When <P is nearly equal to a, the formula may be confider- 



ably 



