1876.] On Differential Equations of the First Order. 337 



" On the Theory of the Solution of a System of Simultaneous 

 Non- linear Partial Differential Equations of the First Order." 

 By E. J. Nanson. Communicated by Professor Caylev,, 

 F.R.S. Received June 5, read June 17, 1875*. 



Given an equation of the form 



z=$(x v x 2 , . . . x n+r , a v a 2 , . . . a n a r+l ), 



we obtain by differentiation with respect to each of the n-\-r variables 

 n + r equations, together with the original equation n + r-fl equations, 

 from which, eliminating the r-fl constants, we have a system of n non- 

 linear partial differential equations. 



Conversely, given a system of n non-linear partial differential equations 

 with n-\-r independent variables, if there exists an equation 



z=<j>(x v cc 2 , . . . cc n+r , a v a 2 , . . . a r , a r+1 ) 



with ?*+l constants, giving rise, as above, to the given system of n equa- 

 tions, then this is the " complete primitive " of the given system. 



Starting with such a system of partial differential equations, it is in 

 the present paper proposed to determine the conditions which must be 

 satisfied in order that the system may admit of a complete primitive, and 

 also to examine what kind of solution, if any, exists when the conditions 

 above referred to are not satisfied. 



The late Professor Boole has given an elegant method of treating a 

 system of linear partial differential equations of the first order : but I am 

 not aware that any one has considered the case of a non-linear system. 



Let us begin with the case in which the dependent variable z is not 

 explicitly involved in the proposed system, which can therefore be pre- 

 sented in the form 



/ 2 OV ^ 2 > . -.aWn P V P V ■ ■ -Pn+r)=0 



Jn\&ii x 2 i ' • ' x n+n Pii Pv • • 'Pn+r) — U, 



(i) 



where p 1 =— , p 2 = — , &c. . . ., the number of equations being n, and 



the number of independent variables being n + r. It is assumed that 

 fvU •••/»> considered as functions of x v oo„ ...x n+r , p v p 2 , ...p n+r , 

 are mutually independent; i£f v f 2 , ...f n were not mutually independent, 

 then, provided the given system were a consistent one, we could replace 

 it by a new system containing a less number of equations. 



Now the existence of a solution involves the supposition that values of 

 PvPs "-Pn+r can be found which will satisfy the equations (i) and at 



* See Proc. Roy. Soc. vol. xxiii. p. 510. 



