Limits of Oblateness of a Rotating Planet. 701 



The pressure due to any canal is 



§dp = $p(Xdai + Ydi/ + Zdz) . . . . (1) 



taken between the ends o£ the canal. 



Now if this cannot be integrated without expressing p, X, 

 Y, and Z in terms of x, by the equation to the canal, we 

 shall have 



rfp=P=A-/* , (2) 



I. 



'0 



in which f will differ for each canal and the pressure be 

 dependent on the form of the canal. 

 But if 



pfXdx+Ydy+Ztk) 



be an exact differential, then its integral between limits can 

 be expressed solely as a function of those limits; or 



P=^i,yi J ^)-^ , o,yo J ^o) 5 ■ • . (3) 

 and will be the same whatever the form of the canal. In 

 this case equilibrium will be possible. 



Secondly, the values of the integral for all canals drawn 

 from the point within to different points of the surface 

 must be the same, since all these must exert the same pressure 

 on the point. That is 



ir(*i»!fu -0 -^fao, </o, %) = ^0' 2 , y* z 2 ) — <M<% Vw ~o) 



or f (x 2 ,y 2 , z 3 ) =^(.i'i,^/i, zj . 



Therefore the variation of ^P = at the surface, 



or Xdx + Ydy + Zdz there =0, .... (4) 



which indicates that the whole force must be perpendicular 

 to that surface. These two conditions completely satisfy the 

 problem. 



4. Now if the force be any function of the distance, say 

 <j>r, then 



X=-6r-, Y = -d>r^, Z=-6r~, 

 r r r r r 



and since r 2 =x 2 +y 2 + z 2 



rdr=xdx +ydy + zdz, 



and Hdx + Ydy + Zdz — — <pr.dv, 



or dp = (Xdx + Ydy + Zdz) 



is an exact differential if p is a function of 



Xdx+Ydy+Zdz. 



