Mr. Ivory's Solution of' a Geodetical Problem. 37 



when e is evanescent, the criterion belongs to the great cir- 

 cles, which are therefore the lines of shortest distance. In 

 the other extreme case, when the spheroid flattens into a plane, 

 the cosine of the latitude is the rectilinear distance from the 

 pole, and the criterion belongs to any straight line drawn in 

 the plane. 



As the last equation is general for all points of the geodeti- 

 cal line, it will be true when applied to the initial point of the 

 measurement ; and thus we get 



Cos X sin p = ^7I=r • 



Consequently, 



\/T+?. cos $ If 



Cos A. sin u. = 'LL 



t~ v 



From this equation we obtain 



, ^/l+e* sin*i£ cos X sin p d if< 



^~ s/l T e l X cos ^ V c °s' ^-cos-i X sinV : 



and again, by substituting this value, we get 



j d 4, cos ^ +/\ -\-e* sin- 4 1 



^/cos^ i£ — cos' X sin- ft 



It will now be proper to introduce two new quantities, <$>' 

 and 5', connected with f and s by these equations, viz. 



ds 



ds — / . . , ■ tt7 • 



V 1-j-e' sin 2 ■v£ 



And when these new quantities are substituted, the foregoing 

 equations will become 



, cos X cos (td$ 



' cos i£ yy/cos' \£ — cos' X sin- p. ' 



d\£ cos i£ 



(B) 



•v/cos' \f — cos' X si n-ft 



In these last equations there are now no traces of the 

 spheroid. They express the relations of the parts of a sphe- 

 rical triangle; the sides being s', 90° — X, 90° — vj/; and $', ju.', 

 are the angles opposite to s' and 90°—^. The third angle of 

 the triangle is //, or the azimuth at the end of the geodetical 

 line; and it is determined by the equation 

 Cos \J/ sin |x'= cos A sin p.. 



The equations (A) .show in what manner the quantities <p 

 and 5, which belong to the spheroid, depend upon the like 

 quantities $ and i* on the surface of the sphere. 



In 



