1863.] 423 



All these methods possess, with reference to the requirements of 

 the actual case, a supeTfluous generality. A single common integral 

 of the system is all that is required. 



Now the chief result to be established in this paper is the following. 



If, with Jacobi, according to his second method, we suppose one 

 integral of the un transformed first partial differential equation to he 

 found, if by means of this we construct according to a certain type a 

 new partial diiferential equation, if to the system thus increased we 

 apply the process of my former paper, continually deriving new par- 

 tial differential equations until, no more arising, the system is com- 

 plete, then, under a certain condition hereafter to be explained, a 

 common integral of all the equations of the complete system, and 

 therefore of the original system which is contained in it, may be 

 found by the integration of a single differential equation susceptible 

 of being made integrable by means of a factor. 



But if the condition referred to is not satisfied, a new integral of the 

 first partial differential equation must be found and the process re- 

 peated, with the certainty that sooner or later it will succeed. 



As soon, then, as the condition is satisfied, a solution not, as by 

 Jacobi's methods, first of two of the partial differential equations of 

 the given system, then of three, and so on, but of all at once, is ob- 

 tained ; and this solution is obtained, not as in my former process, 

 by the solution of a system of equations reducible to the exact differ- 

 ential form, but by that of a single differential equation reducible to 

 that form. 



The condition in question is grounded on the theoretical connexion 

 which exists between the process of derivation of partial differential 

 equations involved in my former method, and the process of deriva- 

 tion of integrals involved in Jacobi's methods. In the actual pro- 

 blem, and in virtue of the peculiar form of the partial differential equa- 

 tion given, these two processes are coordinate. If each be carried to 

 its utmost extent, then to each new partial differential equation arising 

 from the one will correspond a new integral (of the first partial dif- 

 ferential equation) arising from the other. The theory now to be 

 developed is founded upon the inquiry whether it is possible to satisfy 

 the completed system of partial differential equations by a function of 

 the completed system of the Jacobian integrals, i. e. to determine a 

 common integral of the completed series of equations as a function of 



2 H 2 



