OP ARTS AND SCIENCES. 229 



for, along the normal, we have 



Also 



ybx x8y zSx x8z 



x8x j^ ySy ^^ zBz _ x^Ba ^^ y-bb _, z^5c 



(x^ . y^ . z^\ const. 

 .'. const. = 2. 



Let F denote the potential due to the shell on its exterior. Then 



d V =k { Qxdx 4- Q^ydy -f- Qzdz} . 



Hence for all points on the ellipsoid whose axes are 2a, 25, 2c, 



F- n = I {Q,x^ + Q^' + ^,.^ - ^„< - Q^,' - ^,v}, 



where V and F^ are the potentials at xyz and x^y^z^, both points being 

 on the ellipsoidal surface of which the axes are 2 a, 2 b, 2 c, 

 Now let 



a be 



so that 



, ada 7 Jt?6 , cdc 



x^dx^ — — g- , yo^^o — "3^ > ^o^^o — ~^ ' 



Then 



a 



V, = IJiQ^ada + Q.bdb + ^,0^4 = I Q, 



where 



dt 



Q 



J V(a^ 



o 



