422 [Jan. 22, 



he shows that, in virtue of the form of the equations and the relation 

 which connects the first and second of them, other integrals of the 

 first equation may be derived by mere processes of differentiation 

 from the integral already found ; and he shows how, by means of such 

 integrals of the first equation, a common integral of the first and 

 second equations of the system may be found. This common inte- 

 gral is a function of the known integral and certain variables, and 

 its form is obtained by the solution of a differential equation between 

 two variables a differential equation which is in general non-linear, 

 and of an order equal to the total number of integrals previously found. 



An integral of the first two equations of the given system having 

 been obtained, Jacobi shows that by a second process of derivation, 

 followed by the solution of a second differential equation, an inte- 

 gral which will satisfy simultaneously the first three equations of the 

 system may be found ; and thus he proceeds by alternate processes 

 of derivation and integration till an integral satisfying all the equa- 

 tions of the given system together is obtained. In these alternations, 

 it is the function of the processes of derivation to give new integrals 

 of the equations already satisfied ; it is the function of the processes 

 of integration to determine the functional forms by which the remain- 

 ing equations may in their turn be satisfied. 



Jacobi' s second method does not require a preliminary trans- 

 formation of the equations ; but the process of derivation, by which 

 from an integral of the first equation other integrals are derived by 

 virtue of the relation connecting the first and second equations, is 

 carried further than in his first method. It is indeed carried on 

 until no new integrals arise. The difference of result is, that the 

 common integral of the first and second partial differential equations 

 is determined as a function solely of the integrals known, and not as a 

 mixed function of integrals and variables. But its form is determined, 

 as before, by the solution of a differential equation. All the subse- 

 quent processes of derivation and integration are of a similar nature. 



On the other hand, the method of my former paper applied to the 

 same problem leads, by a certain process of derivation, to a system of 

 ordinary differential equations equal in number to the number of 

 possible integrals, and, without being individually exact, susceptible 

 of combination into exact differential equations. The integration of 

 these would give all the common integrals of the given system. 



