38 Mr. Ivory's Solution of a Geodetical Problem. 



In the spherical triangle two parts only are known, namely, 

 the side 90° — X, and the angle ft : but it is easy to find the side 

 s' from the length of s, which is known by actual measure- 

 ment. Since e 9 - is a very small fraction, we may, in the for- 

 mula 



ds 



neglect the variation of the latitude, and suppose 4> = X; then 



s — ^/i+s t sin*XJ 



which is an approximation sufficiently near. To obtain an- 

 other approximation, put cos 4/ = cos X in the equations (B), 

 then 



cosjtt 

 ty s= X + COS [X, X s'= X + COS JU- X 5. 



Wherefore, at the middle point of the geodetical line, we have 



, COS it 



^ = X + — f^s; 



and if we suppose that, in the foregoing formula, -\> does not 

 vary from the mean value, we shall get 



s = 



/ . cos u . 



V l+eisin'2(xH -s) 



This last value is now more than sufficient for any practical 

 purpose. 



Having found s', the other parts of the spherical triangle 

 that remain unknown, are to be computed by the rules of 

 spherical trigonometry. By this means we obtain, 



Sin $ = sin (X+8x) = sin X cos s' + cos X cos ^ sin 5' ; 



. , cos A. sin ft 



bin u! = -r-r ; 



r cos (,X-)-S X) ' 



by which formulae, the difference of latitude 8x, and the azi- 

 muth jot,', will be found. 



And, again, we have further, 



sin ft sin s' 



Sin <p'-. 



cos (\ )- «X) 



and, $' being found, it is manifest from what has already been 

 said that we shall obtain <J>, or the true difference of longi- 

 tude in the spheroid, by the formula 



= , x s/^ffl. 



* P " j.-)- e i 



In practice it will be most convenient to reduce the expres- 

 sions 



