284 On the Integration of Partial Differential Equations. 



differential equation of lower order in virtue of which the original 

 equation is satisfied, is that the other equations, which do possess an 

 intermediary integral, are a class apart and have been considered 

 elsewhere.* 



In order to solve a given equation, a system of subsidiary equations 

 is constructed; and the system is made up of two parts. One of these 

 parts is a set of simultaneous partial differential equations in two 

 independent variables and a number of dependent variables, this 

 number being one more than the number of the equations. An 

 integral equivalent of this part accordingly contains an undetermined 

 quantity. The other of the parts is a set of equations in a single 

 independent variable ; it appears that the set of equations in the 

 second part can be consistently satisfied by a determination of the 

 unknown quantity emerging from the first part. 



In particular, the equations represented by 



b, c, f, g, h, l } m, n, v, x, y, z) — 0, 



where x, y, z are the independent variables, v is the dependent 

 variable, Z, m, n are its first derivatives, and a, b, c, f, g, In are its 

 second derivatives, are found to divide themselves into two distinct 

 classes according as the equation 



oa Co cc on, eg of 



is resolvable or is not resolvable into two equations linear in £, 17, 

 When this equation, called the characteristic invariant on account of 

 an invariantive property which it is proved to possess, is resolvable 

 into two linear equations, the process of integration of the sub- 

 sidiary equations is much simplified. , 



The first of the three sections, into which the paper is divided, 

 deals with the general theory, and indicates a method whereby 

 subsidiary equations for an equation F = of any degree in the 

 derivatives of the second order can be constructed. If integrable 

 combinations of the subsidiary system are not obtainable, an exten- 

 sion of the method shows how equations of higher order (when 

 obtainable) can be deduced and associated with the given equation. 



The second of the three sections deals with those equations of 

 which the characteristic invariant is resolvable ; and some examples 

 are given, alike of equations for which the integration of the initial 

 subsidiary system is possible, and of equations for which the ex- 

 tended method must be used. 



The third of the three sections deals with those equations of which 

 the characteristic invariant is irresolvable. Of such equations the 



* ' Carab. Phil. Trans,,' vol. 16, pp. 191-218. 



