14 [March 6, 



number of the integrals, and capable of expression in the form of 

 exact differentials. 



I will confine my observations to the case of n 2 differential equa- 

 tions connecting n variables. The general theory will be seen in the 

 particular one. 



1* The solution of n 2 differential equations of the first order 

 connecting n variables may be reduced to the solution of a system 

 of 2 linear partial differential equations. To deduce these, let P=c 

 be any integral of the given system, and suppose x lt x. 2 . . x n the 

 variables, then from 



d^dxdx 4-^5^=0 



dx^ " dx 2 dx n 



eliminate by means of the given system n 2 of the differentials, and 

 equate to the coefficients of the two remaining and independent 

 ones. 



2. Let the two partial differential equations thus formed be 



dx l ax 2 dx n 



dP dP J~P 



T> t* *- I T W* i T> U JL /-w f-r -r \ 



1 J * 2 7 * * I n -j ~~ V > , \ * / 



then representing 



A d i A d J_A^V 



n 



the equations become 



A 1 P=0, A 2 P=0 ............... (1) 



Form now the equation A x A 2 P A 2 A x P=0, or as it is permitted to 

 express it, 



(A 1 A 2 -^A 2 A l )P=0 ............ ...... (2) 



This will also prove a linear partial differential equation of the first 



dP 

 order ; and if from it by means of (I.) and (II.) we eliminate - - 



dXn-l 



and , we shall obtain an equation of the form 

 dx n 



c dP dP + C n _ 2 -/L_ =0 ......... (HI.) 



l am i dx 2 dxn-z 



This we shall represent by A 3 P=0. The equations (I.) and (II.) 



