1863.] 421 



in the case under consideration that auxiliary system consisted of the 

 dynamical equations themselves. Jacobi's new methods do not 

 require the preliminary integration of the auxiliary system. They re- 

 quire, instead of this, the solution of certain systems of simultaneous 

 linear partial differential equations. To this object, therefore, the 

 method developed in my recent paper on " Simultaneous Differential 

 Equations" (Philosophical Transactions for 1862) might be applied. 

 But the systems of equations in question are of a peculiar form. 

 They admit, in consequence of this, of a peculiar analysis. And 

 Jacobi's methods of solving them are in fact different from mine, 

 though connected with it by remarkable relations. He does indeed 

 refer to the general problem of the solution of simultaneous partial 

 differential equations, and this in language which does not even sup- 

 pose the condition of linearity. He says, " Non ego hie immorabor 

 qusestioni generali quando et quomodo duabus compluribusve aequa- 

 tionibus differentialibus partialibus una eademque functione satisfied 

 possit, sed ad casum propositum investigationem restringam. Quippe 

 quo prseclaris uti licet artificiis ad integrationem expediendam com- 

 modis." But he does not, as far as I have been able to discover, 

 discuss any systems of equations more general than those which 

 arise in the immediate problem before him. 



It is only very lately that I have come to understand the nature of 

 the relation between the general method of solving simultaneous 

 partial differential equations, published in my recent memoir, and the 

 particular methods of Jacobi. But in arriving at this knowledge I 

 have been led to perceive how, by a combination of my own method 

 with one of those of Jacobi, the problem may be solved in a new and 

 perhaps better, certainly a remarkable way. This new way forms 

 the subject of the present paper*. Before proceeding to explain it, 

 it will be necessary to describe Jacobi's methods, to refer to my own 

 already published, aud to point out the nature of the connexion 

 between them. 



The system of linear partial differential equations being given, and 

 it being required to find a simultaneous solution of them, Jacobi, 

 according to his first method, transforms these equations by a change 

 of variables ; he directs that an integral of the first equation be found ; 



* It was stated by me, but without demonstration, at the Meeting of the British 

 Association in Cambridge in October of the present year (1862). 



VOL. XII. 2 H 



