284 On a Difficulty in the Theory of the Attraction of Spher oid 



equal to <p x PR; call this attraction A, and let a be the radius 

 of the sphere. 



Since r 11 is the difference of r and r\ we have r 2 + r' 2 —r ,n 

 = 2r r' = 2 PR x P'Q', and multiplying by £4 we find l <p ?' 2 

 + 1^*-^$,** = 2a$xPR, that is V + V'-V" = 2 a A. 

 The same thing is true for any other three pyramids similarly 

 related to each other, throughout the whole extent of the 

 three solids which are exhausted by them at the same time; 

 and hence, if we now denote by V, V 7 , V", the whole values of 

 the function V for the three solids, and by A the whole attrac- 

 tion of the first of them parallel to PQ on a point at P, we 

 shall still have V + V'-V = 2 aA.^ 



To express this general theorem in the notation of Laplace, 

 we have merely to observe that the attraction A is synony- 

 mous with — (—7- ], and that the quantity V for the sphere 

 is equal to \nt a 2 . Substituting these values, we find 

 V + 2a(^) = _$™2 + V''; 



an exact equation, differing from the approximate one of La- 

 place only in containing the quantity V", and totally indepen- 

 dent of the nature of the surface or the magnitude of the sphere, 

 the only things supposed being that all the lines drawn from 

 P meet the surface again but once, and that no part of it 

 passes beyond a plane through P at right angles to PQ. 



With respect to the limit of the quantity V", it is obvious 

 that if a hemisphere be described from P" as a centre with a 

 radius equal to the greatest difference 8 between the lines 

 P/7, P'/»', the solid P"p u will lie wholly within this hemisphere, 

 a,nd consequently V" will be less than the value of V for the 

 hemisphere, that is, less than W 2 ; for here all the little py- 

 ramids from the centre have the same length 8, and their 

 bases are spread over the hemispherical surface; wherefore 

 V" = 2 7T x \ 8 2 = "n 8 3 . All this is independent of anything 

 but the suppositions just mentioned. 



If now PQ be supposed to be a spheroid of any sort, slightly 

 differing from the sphere P'Q', and such that the line PQ, 

 perpendicular to the surface at P, passes nearly through the 

 centre, then all the differences, of which 8 is the greatest, be- 

 ing of the first order, the quantity V", which is less than tt8 9 , 

 will be of the second order; and therefore neglecting, as 

 Laplace has done, the quantities of that order, we get the 

 theorem in question. 



It may be well to apply the general theorem to the simple 



