a Proposition in the Mccanique Celeste. 85 



differentials relatively to the arcs 9 and ^}/, which equation, as 

 it is well known, need not be here transcribed. This property 

 is owing to the combinations of 6 and 4/ which enter into the 

 composition of these coefficients ; and as the same combina- 

 tions remain unchanged in the integrals, Y,, these integrals 

 necessarily satisfy the same equation of partial differentials. 

 Thus the peculiar nature of the integrals Y, does not depend 

 upon the function / (9', -^i'), but is derived from the coeffi- 

 cients Pj . 



It is obvious that the coefficients Pj are every one suscep- 

 tible of one form only : and hence the several integrals, Yj , 

 taken between the extreme limits of the variable arcs, are sus- 

 ceptible of one value only. 



When a is less than 1, the foregoing value of X is ex- 

 pressed by a converging series, single in its form. If we put 

 X' for what X becomes when « = 1, and further suppose that 

 the integrals, Y, , form a converging series, we shall have 



X' = Yo+ Yi + Y^ Y, 



The value of X' is therefore a series, single in its form, and 

 such as M. Poisson has determined. But, in what has been 

 said, there is not a word to prove that X' =f{^, vj/), or that 

 X' will be found by substituting in/ (9', •]>'), the initial arcs 

 fl, \I/, for the variable arcs 9', v^'. The proof of this is the se- 

 cond part of M. Poisson's investigation ; and it is here that 

 the difficulty lies. Those who hold that his investigation is 

 rigorous, must admit the truth of the resulting theorem ; those 

 who are content with results, cannot repose confidence on 

 better authority ; those who seek unexceptionable evidence in 

 the mathematics, may find reason to demur at some parts of 

 the proof. As Mr. Pratt has not touched on this point, it is 

 not necessary to notice it further. 



2ndly. Let us now make a particular supposition respecting 

 f{^\'^\ namely, that it is a rational function in finite terms 

 of the three quantities, cos 9', sin 9', cos;}/', sin 9', sin v|/'; which 

 will be verified both when/(9', \J/') is actually a finite func- 

 tion of the three quantities, and when it is a converging series 

 of such functions. According to what is usually taught 

 f{P', \I'') andy(9, v)/) may, each, be expanded at least into 

 one determinate series, viz. 



/{^',^>') = Zo' + z/ + z^' ... z/ ... , 

 /(9, vj/) = Zo + z, + z^... z, ... , 



all the terms of the developments satisfying the fundamental 

 equation in partial diflerentials. If we now substitute the 



