94r Letter from Mr. Ivory to the Editors. 



^ ^ is jfinite in the whole extent of the integral, which 

 -/!-;' 



limits the method to functions, or portions of functions, that 

 can be expressed rationally in terms of three rectangular co- 

 ordinates. M. Poisson confines his demonstration to func- 

 tions that are finite within the limits of integration, and he 

 uses the expression /b«c//o?2S entierement arbitraires. This is 

 rather vague and indefinite ; but if he will allow that he means 

 no more than rational expressions of three rectangular co- 

 ordinates, either in a finite form or in a converging series, 

 every ground of dispute will be removed. 



Nothing can be more unprecise or unsatisfactory than what 

 we can gather in different authors respecting the extent of 

 this analytical method. In the Mec. Celeste, livr. 3™% the ut- 

 most generality seems to be aimed at. In a Memoir published 

 by the same author in 1816*, and afterwards in the 11th book 

 of the Mec. Celeste, the demonstration of the fundamental 

 equation is modified so as to fall far within the real limits of 

 the method. Professor Airy has since republished this last 

 demonstration of Laplacef, with some explanations ; and, as 

 he takes no notice of its limited application, we must infer 

 that he conceives it to be general. The Professor seems to 

 have been led into this peccadillo against geometrical rigour 

 by his deference to the highest authorities. The process of 

 Lagrange:}: is not liable to objection; but it is imperfect, inas- 

 much as he has not shown that all the operations which it 

 necessarily supposes, whether they finally stop or run on in- 

 definitely, may be performed without introducing any small 

 divisors that would vitiate the reasoning. Had this been ac- 

 complished, the nature and extent of the method would have 

 been fixed with precision. 



I cannot quit this subject without remarking the strange- 

 ness of this dispute. Whatever general words are employed, 

 or whatever undefined symbols are used, yet, in all the appli- 

 cations of the calculus, these words or symbols stand for no- 

 thing but rational expressions of three rectangular coordinates. 



I remain. Gentlemen, 



Your obedient servant, 

 July 16, 1827. James Ivory. 



* Mcmoircs dc P Acad, dcs Sciences. 



\ Transactions of the (^^ambridfje Philosophical Society. 



X 15"" cahicr dit Journal dc P Ecole Po/j/l. 



XIX. Collections 



