396 Prof. A. Gray mi the Attraction of 



element ds. If now we use the transformation already 

 employed above, that is, express the coordinates of A and P 

 in terms of the coordinates of their corresponding points A! 

 and P : , A 7 on the confocal, P x on the given shell, and sub- 

 stitute for ds its proper value in terms of the area ds' of the 

 corresponding element on the confocal as given by (11), and 

 use the theorem (10), the transformed equation is 



-b=i/o , — — = p Q I „ ds' . (j3) 



in which the integral is now taken over the confocal. The 

 only variable factor is now ds' cos @'/r 2 , and it is well known 

 that for the complete confocal shell 



J 



rfs' cos 



= 4tt 



since the point P x is within it. Hence, if V be the potential 

 at P, (13) becomes 



i? dY > dk ahc ni . 



dn ^ k x /(a :i + u){d 2 -\-u){c 2 + u) 



where dn is an infinitesimal step outward along the normal at 

 P to the confocal. But by the equation of the confocal [see 

 also § 29] 



dn = h - du 



and (11) becomes 



7xt j dkdu . 



— d\=irpabc ,. (151 



\/(a 2 + u){b 2 + u){u 2 + u) 



The potential V of the homoeoid at P, and at any other 

 point of the confocal surface, is thus given by 



Y = irpabcdk\ — = = — =, . (IQ) 



where u is now supposed to vary in value as we go from con- 

 focal to confocal outward from P in the integration : the 

 confocal from which the integration starts is that on which P 

 lies, and A/ is the positive root of (7) regarded as a cubic 

 in u. The value of the expression on the right is — V^ -fVj 

 and as V^ =0, we have (16). 



17. We may proceed in precisely the same way when P 

 is within the homoeoid. A confocal ellipsoidal surface of 



