Royal Society. 391 



" On Differential Transformation and the Reversion of Serieses." 

 By J. J. Sylvester, Esq., F.R.S. 



With a view to its publication in the Proceedings of the Society, I 

 take occasion to communicate the result of my investigations, as far as 

 they have yet extended, into the general theory of differential trans- 

 formations, containing a complete and general solution of the import- 

 ant problem of expanding a given partial differential coefficient of a 

 function in respect of one system of independent variables in terms 

 of the partial differential coefficients thereof, in respect to a second 

 system of independent variables, each respectively given as explicit 

 functions of the first set. 



This question may be shown to be exactly coincident with that 

 of the reversion of simultaneous serieses proposed by Jacobi, which 

 may be thus stated: given (w+1) quantities, each expressed by 

 rational infinite serieses as functions of n others ; required to express 

 any one of the first set in a rational infinite series in terms of the 

 other n of the same set. This question has only been resolved by 

 Jacobi for a particular case ; the result hereunder given for the trans- 

 formation of differential coefficients contains the solution of the gene- 

 ral question. My method of investigation is entirely different from 

 that adopted by the great Jacobi, and I hope in a short time to be 

 able to lay it in a complete form before the Society, and probably to 

 add a solution of the still more general question comprising the re- 

 version of serieses as a particular case, viz. the question of express- 

 ing any one of n quantities connected by w equations in terms of 

 any (n — m) others of the same. 



Let there be any number of variables, say u, v, w, of which x, y, z, 

 § are given functions, it is required to expand 



(AY (A\ e ( d \ h 



\dx) \dy) \e 



<dz) 



in terms of the partial differential coefficients of $, x, y, z in respect 

 of u, v, w. 



Form the determinant 



dx dx dx 



du dv dw' 



dy dy dy 



du dv dw' 



dz dz dz 



du dv dw' 

 which call J. 



The required expansion will contain in each term an integer nume- 

 rical coefficient, a power of -, one factor of the form 



(AY (AY ( d )\ 



\du) \dv) \dwj 



and other factors of the form 



