Ellipsoidal Shells and of Solid Ellipsoids. 397 



equation 



%-£- = k (17) 



a- — u 



is described through P and the corresponding point P x on the 



shell is taken as before. P is joined to the point A on the 



shell and P t to the corresponding point A! on the confocal. 



The angle O is as before between AP and the normal at P. 



and 6' between P X A' and the normal at A'; also AP=A'P 1 =r. 



If p , p be the lengths of the perpendiculars from the centre 



on the tangent planes at P and A', we have as before 



p' cos o =p cos 0'. 



Precisely as in the former case we get for the attraction on a 



unit particle at P 



^ die abc fcos<9' 



F = \p~-r = Pa 1 — —as . . (lb) 



2H k ' V (a 2 - u) (lr - u) (c 2 -uf° J r- ' v ; 



where the integral is to be taken over the confocal. The 

 integral is the total solid angle at P 2 subtended by the con- 

 focal. and as P x is outside that surface, the solid angle is 

 zero. Hence the force is zero at every point within the 

 shell, and the potential is there uniform. Thus the problem 

 of the attraction of a homoeoid is completely solved. 



"We have, in the result stated in (13), the curious theorem 



that the value of F at P is, to a constant factor, equal to the 



potential produced, at any point internal to itself, by a 



uniform magnetic shell coinciding with the confocal surface. 



The strength of this shell is proportional to the length of the 



perpendicular from the centre on the tangent plane to the 



confocal at the point P, and therefore varies with the position 



of P on the surface, as does the length of that perpendicular. 



A similar theorem holds for a distribution upon any surface 



whatever, which maintains that surface at uniform potential. 



The potential which that distribution produces at an external 



point P is equal to the potential which a magnetic shell, 



coinciding with the equipotentiai surface through P for the 



same distribution, produces at any point internal to itself. 



The strength of the shell varies with the position of P, and 



is inversely proportional to the distance, dn, of P measured 



along the normal to a choseu neighbouring equipotentiai 



surface. This expresses, in fact, the relation (32) below, 



namely, 



d\ . 



an 

 This shows that c is inversely proportional to dn for a constant 

 dV. [See § 27.] 



