Ellipsoidal Shells and of Solid Ellipsoids. 395 



This theorem, it is to be remarked, is not confined to con- 

 focal ellipsoidal surfaces. 



15. Now, imagine drawn from P as vertex a cone of small 

 solid angle deo, intercepting two elements of the homoeoid at 

 A and B : let ds be the area of that at A. The length of the 

 perpendicular from the centre on the tangent plane at A 

 being p, the mass of the element is ^ppdkds/k ; and if c/s' be 

 the element at A' of the surface of the confocal corresponding 

 to ds, that is, containing the points corresponding to those 

 contained in rfs, we know that 



pd*= , ** g - *'<*»'■ • • (11) 



<s/{a 2 + u)(b* + u)(c 2 + u) 



The attraction at right angles to the axis PQ exerted by 

 the element on a unit particle at P is when k — 1 (see foot- 

 note § 6) 



which by (11) and (10) may be written 



, elk a o c cos 6' eh' „ 



ipPa i- -, —-:■-- ._ .. _ - tan c/ . 



k Jl^+^{b 2 + u)^ + u) r- 



In this the only factor which varies from element to element is 



cos0' da' , n 



5 tan r . 



ir 



Now cos 6' ds'/r 2 is the solid angle dco subtended at F 1 by the 

 element d$' at A'. We can exhaust the whole of the homoeoid 

 by means of elements intercepted by small cones drawn from 

 P as vertex ; and to this corresponds precisely an exhaustion 

 of the confocal by small cones radiating from the internal 

 point P 2 as vertex. And clearly for every elementary cone 

 of solid angle do), there exists another in the same plane 

 through PQ, for which the factor just referred to has the 

 same value with opposite sign. Hence in no plane through PQ 

 is there any force perpendicular to PQ on a unit particle at 

 P ; that is the resultant is along PQ. This is Poisson's 

 theorem. 



16. The total force F along PQ (in the direction from P to 



Q) is given by 



-^ . dk C cos 6 ds , H ~ 



F=ip T J^— J-, .... (12) 



in which the expression to be integrated over the homoeoid is 

 the component of attraction along PQ clue to the single 



