340 Mr. E. J. Nansou on Differential 



d<p d(j> d<f> 



/*/ dx. dx\ d& n i r \ 

 f{*, , v „ p . . . x v+n - ^, - ^ . . . — ^ = o («) 



dz dz dz 



Hence (j> considered as a function of the n + r+1 independent variables 

 z v x, x 2 , . ..x n+r must satisfy a system of n partial differential equations 

 of the first order ; and in this system the dependent variable does not 

 appear explicitly. If, then, there be any value of <j> which satisfies the 

 system, it can be found by the method given above ; and provided it in- 

 volve z in its expression, the value of z found from the equation 0=0 

 will be a solution of the original system. If z does not occur in the ex- 

 pression for (p, then the proposed system can have no solution. 



Suppose the system of equations of which (ix) is the type to have a 

 complete primitive of the form 



<j) = \p(z, x v a? 2 , . .. x n+r , a v ci 2 , . . .a r+l )-\-a r+2 



containing the r + 2 arbitrary constants a v a 2 , . . . a r+2 . Then the equa- 

 tion (j)=0 gives us the value of z in the form 



z = B(co v x 2 , ... x n+r , a v a 2 , . . . a r+2 ) (x) 



And it is to be observed that this value appears to contain one more than 

 the number of arbitrary constants indicated by the theory of the genesis 

 of the system (vii) as the proper number. But from the fact that (x) 

 satisfies the system (vii) of n equations, it follows that the constants a v 

 a 2 , . . . a r+2 must be virtually equivalent to r+1 constants only. An in- 

 stance of this occurs in the first example given below. 



The results of the preceding inquiry may be collected into the follow- 

 ing rules. 



Given a system of n non-linear partial differential equations of the first 

 order in w+r independent variables, and in which the dependent variable 

 does not explicitly appear, to find the nature of the possible solution. 



Mules Let the equations by algebraical reduction be brought to the 

 form 



: f W 



and examine whether the condition 



Ui,n= m - dYj + k 7( dFi . dFj .- gS. -^M=o.(xii) 



<% d *i k=i\ d *'n+k dj?n+k dr n+k 'dp n+h J 



is identically satisfied for every pair of the functions I\, ... P„. If it be 

 so, then the system will have a complete primitive of the form 



z =([)(#„ w 2 , ... x n+r , ci v a 2 , . . . « r ) + J, (xiii) 



which may be formed by the method of Boole referred to. But if anv 



