219 



The objections that have been and possibly may be used against 

 diamagnetic polarity, are next considered, and some observations are 

 made on the constitution of the magnetic field. The relation of 

 our present knowledge of diamagnetic action to the theory of Weber, 

 and to Ampere's hypothesis of molecular currents is stated ; and in 

 conclusion, the author dwells briefly upon those points of diamag- 

 netic action wherein the views of M. Matteucci differ from his own. 



The following paper was read : 



"On Differential Transformation and the Reversion of Serieses." 

 By J. J. SYLVESTER, Esq., F.R.S. Received January 25, 

 1855. 



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 (+!) 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- 



