EQUATIONS OF THE FIEST OEDEE. 
449 
If all such prove to be identities, the given system of differential equations admits of n 
integrals, and is reducible to a system of exact differential equations. But if any such 
equation is not an identity, it will constitute a new partial differential equation of the 
form 
And this, combined with the previous ones, will enable us to form a system of r+1 
partial differential equations, in which •> ... - appear each in only one equa- 
^ ^ ^ dXn ClXn+i dXn+r ^ J M 
tion and with coefficient unity. Upon this system let the same process be repeated as 
upon the pre\ious system of r partial differential equations, and so continually repeated 
until we arrive at a final system of partial differential equations such that, if that system 
be represented in the form 
A,P = 0...A,„P=:0, 
the condition 
(a,a_AjA,)p=0 
shall be identically satisfied for every pair. 
Then, the number of such partial differential equations being m, the number of inte- 
grals of the original system of partial differential equations will be m, i. e. it will 
be equal to the number of the original variables diminished by the number of final 
partial differential equations. 
And if by that final system we eliminate m of the differential coefficients from 
dV , dV . , dP 
dx^ ^^~^dX2 dXn+r 
and equate to 0 the coefficients of the remaining differential coefficients, w’e shall have 
a system of n-\-r—m differential equations expressible as exact differential equations for 
the determination of the integrals. 
Actually to determine these, we should endeavour in the first instance to reduce the 
final system of differential equations, as such reduction is theoretically possible, to a system 
of exact differential equations. If the means of doin^ this are not obvious, the method 
of the variation of parameters or the equivalent methods of Prop. III. must be applied. 
Lastly, if the process which consists in the application of the theorem (AAj — AjAi)P=0 
do not stop with the formation of the final system of partial differential equations, but 
lead to algebraic relations among the variables, the given system of differential equa- 
tions will have no integrals properly so called, but it may admit of solutions analogous 
to those the theory of which has been developed by Pfaff, Jacobi, and others for the 
differential equation 
0 . 
