402 Prof. Collins on the Attraction of Ellipsoids 



proposition, the sum of the attractions exerted on P along PP' 

 by the two small pyramids PEFE^F,, PE'F'E'^F', is equal to the 

 sum of the attractions exerted on p along pp' by tlie two corre- 

 sponding small pjTamids pefeJi, ps'f'e'if'i, since the solid angles 

 of the four pyramids are all equal to each other : and since there 

 are obviously as many pairs of pyramids in the double wedge PP'E 

 FE'F' as there are corresponding pairs of pyramids (whose solid <s 

 are also equal to those of the former) in the double wedge pp^ef^f , 

 and as, moreover, each double wedge of A has a corresponding 

 double wedge of «, .•. the whole attraction of a on p is equal to 

 the attraction of A on P along PP' ; but since a is similar to A, 

 the attraction of « Qn.p : attraction of A on C : : jo/>'( = PP') : CC 

 {Principia, Prop. 87, Cor. 1, Book 1) ; and so the attraction of 

 A on P perpendicular to B, which was proved equal to attraction 



iPP' 

 oi a onp, is .'. = Ynni ^ attraction of A on C, which, since A 



and CC are constant, cc |PP', which is the distance of P from B. 



3. The general equation of surfaces of the second order being 

 A{xyz)=A + Bx + Ct/ + 'Dz + 'Eix'^ + kc. =0, the equation of the 

 diametral plane bisecting all chords parallel to the straight line 

 x=-mz and y = nz'\^ known to be mdj^-\-nd <^-\-d_^-=.Q, which, 

 on account of not containing the absolute constant term A, indi- 

 cates that if any straight line ABB' A' cuts two surfaces of the 

 second order whose equations differ only in the constant terms, 

 the intercepts AB, A'B' will be equal, since the chords AA', BB' 

 are bisected in the same point by the diametral plane conjugate 

 to it, which plane is the same for both surfaces. Now the equa- 

 tions of two ellipsoids which are concentric, similar, and simi- 

 larly placed, diifer only in the absolute constant terms, and 

 .•. the intercepts AB, A'B' of any straight line ABB' A' cutting 

 two such ellipsoids are equal ; and hence it follows that a shell 

 or couch bounded by two concentric, similar, and simdary placed 

 ellipsoidal surfaces, exerts no attraction on a point P situated 

 anywhere within it, or on its interior surface. See Airy's Tract 

 on the Figure of the Earth, Prop. 12, or the Principia, Prop. 70, 

 and Prop. 91, Cor. 3, Book 1. 



3. When P is within the ellipsoid A, we have then only to 

 describe through P another ellipsoid A', concentric, similar, and 

 similarly situated to A ; and since the shell between A and A' 

 exerts no attraction on P, as was just proved, .". the whole attrac- 

 tion of A on P is the same as that of A' on P : then, as in 1, 

 draw PP' a chord of A' perpendicular to B, and through P and 

 P' draw planes parallel to B cutting the principal axis CC per- 

 pendicularly in p and />', and then describe through p andy 

 another ellipsoid «, concentric, similar, and similarly placed to 

 A or A'; and by 1, the attraction of A (or A') on P perpendi- 



