328 Mr. Ivory's Remarks on M. Poissou's Memoir. 



sin 9 sin vl/, which possess the supposed property. In all other 

 cases the value of the above expression is indeterminate, and 

 the demonstration of M. Poisson's formula, or, which is the 

 same thing, of Laplace's fundamental equation in partial dif- 

 ferentials, ceases to be exact. 



Mr. Professor Airy, in a short jiaper read to the Cambridge 

 Philosophical Society in May last, and printed in their Trans- 

 actions, has treated of this subject ; and he advances rather a 

 singular opinion. He agrees with me that the method of La- 

 place must be limited to a particular class of spheroids; and 

 he claims the honour of having first placed the matter in its 

 true light. But he attempts to show that the fundamental 

 equation, Mcc. Celeste, liv. iii. No. 10, is exactly demon- 

 strated. Now admitting that the equation in question is ac- 

 curately and numerically proved, it seems impossible to deny 

 that the series of terms deduced from it, is numericallij equal 

 to the distance between the surfaces of the sphere and sphe- 

 roid. I have always contended that the fault lay in the sup- 

 posed generality of the equation, which is true only in a par- 

 ticular class of spheroids. On the other hand MM. Laplace and 

 Poisson have upheld the universality of the equation by new 

 proofs, of which I have here had occasion to speak. In my 

 view the theory is freed from its difficulties, and becomes 

 satisfactory, although stript of its high pretensions to gene- 

 rality. Mr. Professor Aii'y, by supporting the fundamental 

 equation without i-estricting it, and at the same time denying 

 the unavoidable consequence, has only introduced new incon- 

 sistencies, and embroiled, with new difficulties, a subject very 

 seducing by its analytical elegance, but very perplexing when 

 we resolutely seek to exhibit to the understanding a rational 

 account of its principles. 



In examining the theory of Laplace, the want of rigour in 

 the analysis could hardly escape detection ; and in a subject of 

 so great interest and difficulty, it seemed requisite to scruti- 

 nize and clear up every doubtful point. But the nature of the 

 analysis will become a consideration of only secondary im- 

 portance, if it shall appear that there are defects in the first 

 principles, or in the conditions of efjuilibrium. In the pro- 

 blem of the figure of a planet in a fluid state, thei'e are too 

 different cases ; for we may suppose it to consist of only one 

 homogeneous fluid, or of sevei'al fluids arranged in strata 

 varying in density from the centre to the surface. If the first 

 case were solved, the theory of equilibrium of which we are 

 in possession, would be sufficient for investigating the second 

 case. But the present theory fails in the equilibrium of a ho- 

 mogeneous planet. I have found that the equilibrium cannot 



take 



