82 Mr. Ivory om the Equilibrium of ajiuid Planet. 



Const. = fiFdx + QcZj/ + R dr.) + -^ {^^ + f) 



+ r(?<dx + Q'd7/+ R'dz). 



Now the least attention to the nature of this equation will 

 show that the attraction of the matter without the level sur- 

 face is entirely independent of the rest of the equation. As 

 there is no definite relation between P, Q, R and P', Q', R', 

 these functions denoting the attractive forces of different 

 quantities of matter, a level surface will have no determinate 

 figure, unless we divide its equation into two separate parts 

 containing the unrelated quantities, viz. 



Const. = r{Pdx+Qdi/+Rd%)+ ^{x^+f) (2.) 



Const. :=r{P'dx + Q'di^ + R'dz) (3.) 



Such are the equations of equilibrium of a homogeneous 

 planet supposed fluid : but it will be more convenient to use 

 the two following, of which equation (1.) is the sum of (2.) 

 and (3.), 



Const. = fiXdx +Ydi/+Zdz) + -^ i^^+f) — (1-) 



Const. = nF'dx -{-Q'd 7/+ R'dz) (3.) 



Now these two equations are the same with those given in 

 a paper in the Philosophical Transactions for ] 824, and in 

 two subsequent papers written for the purpose of obviating 

 some objections (I had almost said, frivolous objections) of 

 M. Poisson*. The foregoing very simple investigation proves 

 the justness of the solution of the problem in the papers 

 cited ; and at the same time shows the insufficiency and in- 

 correctness of making the figure of equilibrium depend on 

 one equation. (Poisson, Mecanique, p. 550, vol. ii., edit. 2nd.) 



What is said is sufficient for the present purpose : but the 

 subject deserves a more extended discussion ; because, by the 

 procedure here followed, the greatest simplicity and clear- 

 ness are introduced in one of the most perplexed and imper- 

 fect theories, that occur in the system of the universe. 



July 10, 1838. James Ivory. 



* See Phil. Mag., First Series, vol. Ixiii. p. 339; vol. Ixv. p. 241 ; vol. 

 Ixvi. p. 429; vol.lxvii. p. 31, 82, 439.— Edit, 



