* 510 MR. IVORY ON THE EQUILIBRIUM OF A MASS OF 





Because every level surface is similar to the upper surface, and similarly posited about 

 the centre of gravity, and that r and R, as well as r' and R', are radii of the two 

 surfaces in the same straight line, we have 



r = a . R, r' = a . R', 



« being a fraction of unit, which is the same for all the points of the same level sur- 

 face ; wherefore, by substituting the values of r and r', and leaving out the term 



"■'jy- dYd,rlog^X d'^ = R2 «' log I .//- dyd, &'\ 

 which is equal to zero, we get 



f'f = ^/f-drd..^^ 



+ (a - a2) Bjy- dydcd'^ . R' 

 2 J J R'2 



2 — 2 J J R''-2 



Such is the expression of the integral under consideration, the attracted point being 

 the intersection of R, with the level surface of which a R is a radius ; and the value 

 of the integral must be constant at all the points of the same level surface, that is, it 

 must be the same when a is the same in whatever direction R be drawn. 



-7^ is re- 

 duced to its first term, which is constant in every spherical surface concentric with 



