of two Points o? i the Earth's Surface. 31 



gin being in the centre, x being perpendicular to the equator, 

 and y to the fixed meridian, we shall have, by the properties 



of the spheroid, 



A = V 1 — e 2 sin 2 A, 



(\ —e") sin X cos X sin %i cos X cos •vi 



A ' ■* A A 



Further, d s being the element of a curve on the surface of 

 the spheroid, we shall have, 



ds 2 =z dx 2 + dy 2 + dz* ; 

 and if we substitute the differentials of the coordinates, there 

 will result 



, „ (1— e^dX 2 - cos^xd^- 

 ds = — £ + — ^ ' 



This expression is general, whatever be the nature of the 

 curve. In order to find the equation of the geodetical line 

 with the least calculation, I shall make d s and d \J/ only vary ; 



then, d 8 s = d § \f> 



consequently, 



A' z ds 



~ -> , cos-Xd^ ^ , 7 cos 2 X <£■*!< 

 8 S = \p . — : / 8 vl/rf . - . 



The beginning and end of the curvilineal arc being fixed, the 

 nature of the line of shortest distance requires that, 



T cos^xd^ „ -, cos* kd-d' 



d. - — — J — = 0, and, — == c, 



A^ds ' ' A 2 ds 



c being a quantity that remains constant in the whole length 

 of the curve. 



Using now this last equation to exterminate dfy from the 

 general expression of d s 2 , we shall get, 



. /' c"- A 8 (l-e a )dx 



ds V 1 --^ s Tr= — * — : 



and, if we make, 



b 2 - — — 



we shall further obtain, 



d x cos x l 



V 



We likewise have, 



y cos 9 X «^ cos X vo J — sin*X A 



In order to deduce the values of s and \{/ from the foregoing 

 expressions, it would seem to be necessary to integrate 

 them between the initial and final latitudes A.' and A, of which 



the 



/a x cos x 

 a/ b 1 -— sin 5 



A3 



