86 On a Proposition in the Mecanique Celeste. 



series for y" (fl', •^') in the terms of X', we shall obtain simply. 



Yi = p r^ ^ Vif (fl', ^I/') sin h^dh' d ^' 





'^^ P,-Z/sinfl'^fl'J^' = Z,, 



because all the parts ofy(9', \|;'), except only Z, , produce no- 

 thing in the integral. It is, therefore, proved, in this parti- 

 cular supposition respecting^ (9', \I/'), that the developments of 

 X' and_/(65 ■^) are identical in all their terms: and as the first 

 series has been shown to be susceptible of one form only, the 

 same thing must be true of the other series. In order to 

 arrive at this conclusion it was not even necessary to intro- 

 duce the expansion of /(fl', <^') : for, in the case in hand the 

 equation 



X'=/(fl,^I/) 

 is easily deduced by direct integration, as was done many 

 years ago. 



All that we properly know of this theory, because it is all 

 that is strictly proved, is contained in what has been said. 

 The process which M. Poisson has invented for demonstrating 

 the equation 



X'=/(fl,vI/) 



without restriction, is merely an analytical artifice founded on 

 assumptions : it has not been verified by any rigorous investi- 

 gation ; on the contrary, it is opposed to every such investi- 

 gation. 



Mr. Pratt's observations fall to the ground ; since, contrary 

 to what he assumes, every proposed function is susceptible of 

 one development only. The term Y, in the expansion of 

 X', or Zy in the expansion ofy(fl, ^) in the particular case 

 mentioned, is equal to the integral 



n f^ " ^if (fl', ^') sin ^> d^'d ct', 



which has one value only ; because the arcs 9' and 4/' vary 

 between determinate limits ; and, for any assigned values of 

 these arcs, each of the functions P,; and _/ (fl', vj/') is single in 

 its form. 



It seems to be implied in the language used by Mr. Pratt, 

 that Professor Airy was the first who raised objections to this 

 analytical theory propounded by Laplace, and who placed it 

 in the point of view here given of it : but whether this be cor- 

 rect or not will best appear by citing the Professor's own 

 words: — " I conclude with Mr. Ivory, that the theory (of the 



