Mr. Ivory on an Article in the Bulletin des Sciences. 275 



The article in the Bulletiti likewise slightly touches upon 

 Laplace's development, with respect to which M. Poisson has 

 inserted a note in the Con?i. des Terns, 1831. The last re- 

 searches of M. Poisson coincide entirely with what I have all 

 along asserted; namely, that the development is applicable 

 only to rational functions of cos fl, sin 6 cos \J/, sin 6 sin \I/ ; and 

 of this they, in reality, furnish a clear demonstration. He has 

 proved that the development converges, when j/ =y(6', r}/'), 

 has the meaning he affixes to it ; so that taking ti equal to a 

 finite, although perhaps a very great number, we shall have 

 with sufficient exactness, 



^ _ Y(c) + Y") + Y(2) + Y(»). 



Now it is certain that every term of this series is, necessarily 

 by its very structure, a rational function of cos 6, sin fi cos \|/, 

 sin 9 sin ^ ; and therefore the aggregate of any number of 

 such terms is a function of the same kind. And this conclu- 

 sion does not rest upon M. Poisson's demonstrations ; it de- 

 pends upon the convergency of the series, that is, upon its ca- 

 pability of expressing a determinate value. No proposition 

 can possibly be more clearly proved than this, Thaty(6, ^) 

 cannot be developed according to the method of Laplace in a 

 series of converging terms, unless it be equal, either exactly 

 or approximately, to a rational expression of cos 9, sin 9 cos 4/, 

 sin 6 sin \J/. 



Every function/ (9, ^) may be reduced by the ordinary rules 

 of algebra to a rational expression, convergent or not, of cos 9, 

 sin Q cos \|/, sin 6 sin \{/ ; but it is only when the expression con- 

 verges that Laplace's development can be applied with geo- 

 metrical accuracy. The development adds nothing to the to- 

 tality of the reduced expression, nor takes one iota away from 

 it; it merely breaks every coefficient into many parts so as to 

 allow a new arrangement in parcels, every one of which satis- 

 fies the fundamental equation in partial fluxions. 



The subjects treated in this article are of considerable im- 

 portance, and I may occasionally return to them, in order to 

 remove every difficulty. They have given rise to a long con- 

 testation, actively carried on ; and if 1 mistake not, the doc- 

 trine I have advanced has not always engaged the attention 

 only of such authors as I have alluded to, who labour to 

 improve science, but has sometimes been made a handle. 



Sept. 13, 1829. J. Ivory. 



2 N 2 XLI. Notice 



