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XVII. Lettefjo G.B.Airy, Esq., Lucasian Professor of the 

 Mathematics in the University/ of Cambridge. By J. Ivory, 

 Esq. M.A. F.R.S. 

 Sir, 

 TN the letter I did myself the honour to address to you in 

 -^ the last Number of this Journal, I showed that the analysis 

 employed by Laplace in the investigation of the figure of the 

 planets is confined to rational and finite expressions of three 

 rectangular coordinates, or to such functions as can be ex- 

 panded, by converging series, into expressions of the kind 

 mentioned. I proved this from the nature of the development, 

 which appears to me to be the proper foundation of this ana- 

 lytical theory. But it has been usual to rest this doctrine on 

 Laplace's equation in partial differentials, Mec. Celeste, liv. 3^^, 

 No. 10, or on M. Poisson's proposition, Co?in. des Terns, 1829, 

 p. 330 *, which are supposed to be demonstrated with I know 

 not what degree of generality. In order to remove every dif- 

 ficulty, it therefore seems necessary to reconcile the two dif- 

 ferent views that have been taken of this subject, and to prove 

 that the equation of Laplace, and the theorem of M. Poisson, 

 are restricted by the same limitation which the nature of the 

 development requires to be adopted. To accomplish this end 

 is the purpose of the present letter, which I address to you 

 as a person not indifferent to the progress of this branch of 

 science, and, I may add, as the only one in England who 

 seems to have bestowed upon it the least degree of attention. 



The analytical theory, in the view we now take of it, is stated 

 in the proposition of M. Poisson very clearly, and free from 

 any particular application : I shall therefore confine my atten- 

 tion to it. I suppose that y' =f{^', \^') is a function of two 

 variable arcs, and that j/ —f{^, 4') is what y' becomes when 

 the particular values $ and 4/ are assigned to 6' and v}/' : further, 

 put 



p = cos 6' cos 9 + sin S' sin 9 cos (vf/ — <{,'), 



/= >/l-2«iJ + «'- = \/ (1 -«)- + 2a (1-;;), 

 ds = d^' sin Q'di,', 

 a. being supposed less than unit, but tending to it as a limit; 

 then, form the expression, 



1 /- (I -« °)y>/5 

 ^= l^J 7^ ' 



the integral extending to thewhole surfaceof the sphere, or from 

 5' = 0, 4/' = 0, to fl' = IT, \I/' = 27r. Now, in the particular case 



• First imb\h\\{id,JcitirnalclerEcolc Pu/^t. 19"" caliicr, p. 145. 



when 



