90 Colonel A. R. Clarke on the Figure of the Earth. 



This piano is to contain the normal at £,y,z 3 and the point 

 .?• + da } y + dt/, z + dZj which conditions give two other equations 

 in A, B, C ; and eliminating these symbols we have the dif- 

 ferential equation of the required curve expressed by the de- 

 terminant 



'— ** y'-y> *—*> 



dx, dy, dz, 



x y z 



$ P ?' 



A north-and-south line is a particular case of this curve, 

 viz. when #' = 0, y' = 0, z f = co ; then the equation becomes 



a 2 ydx-b 2 xdy = 0, (3) 



of which the integral is 



x a2 = cy b \ 



This is not'a plane curve ; and at each point its direction makes 



a definite angle with the meridian as expressed 



by (1). Let S be any point on the surface of B * N 



the ellipsoid, say in that octant where x, y, z are 



all positive ; let M be a point indefinitely near S 



on the same meridian (1), Na point on the north 



line (3), P a point on the parallel of latitude 



through S, of which, c/> being the latitude, the 



equation is 



2 Jl 



+ |-^cot 2 <£ = (4) 



a 4 ' b 



The differential equation of the meridian is 



— b 2 sin a dx + a 2 coscody = (5) 



And if from this equation, with (2), we determine the ratios of 

 dx, dy, dz, they are found to be proportional to 



— a 2 z cos a : — b 2 z sin to : c 2 (xcoso)+ysina)) . (6) 



And these are proportional to the direction-cosines of SM. 

 So also, getting the ratios of dx, dy, dz from (2) and (3), we 

 find the direction-cosines of SN to be proportional to 



-b 2 xz : -a 2 yz : c 2 [^ 2 y 2 + - 2 x 2 J, .... (7) 



Similarly for the direction-cosines of SP ; they are as 



These enable us to determine the angles between the lines SM, 



