226 PROCEEDINGS OP THE AMERICAN ACADEMY 



XVI. 



THE POTENTIAL OF A SHELL BOUNDED BY CON- 

 FOCAL ELLIPSOIDAL SURFACES. 



By Frank Nelson Cole. 



Presented May 29th, 1883. 



Let there be three confocal ellipsoids, of which the outer two have 

 their bounding surfaces fixed, while that of the inner ellipsoid ap- 

 proaches that of the middle one, the three surfaces always remaining 

 confocal. Let X^^ and Xgg be the respective attractions, resolved 

 parallel to the axis of x, of the inner and middle ellipsoids at any 

 point in the surface of the outer ellipsoid ; and let X^^ and X^^ be 

 the attractions of the outer ellipsoid at the corresponding points of 

 the inner and middle ellipsoids. To calculate the limiting value of 

 X^ — Xj3 as the inner ellipsoid approaches the middle one : — 

 By Ivory's theorem, 



^23 ^ „ ^2 



h'^3 



Also, 

 where 



^"=' J(«= 



^32= 2apa3V3^a3^2» 



dt 



^ + vWToWToW+o ' 



and a, h, c, and x are the semi-axes of an ellipsoid and the x co-ordi- 

 nate of a point, the subscripts denoting the ellipsoid to which the axes 

 belong and on the surface of which the point lies. 

 Hence 



b c 



3 3 



for x^ and x^ being co-ordinates of corresponding points, we have 



