448 
PEOFESSOE BOOLE ON SIMULTANEOUS DIFFEEENTIAL 
Integrate this in the form 
then, treating as functions of the variables before regarded as constant, and 
endeavouring to satisfy the unreduced equations (6.), we obtain, in virtue of the above 
conditions, a system of differential equations equal in number to the system given, but 
containing one variable fewer. The system (13.) by which the forms of are 
here determined, is the Lagrangian auxiliary of the first partial differential equation 
A,P=0 integrated in the other method; and so in each subsequent stage. And with 
respect to the other parts of the process, it obviously makes no difference whether we 
take as the new variables Mj, . .. or Cj, . .. Cp under the condition (necessarily involved in 
the method of the variation of parameters) that they shall after integration be replaced 
by «!,... But it would not have sufficed simply to refer the solution of the final 
system of ordinary differential equations to the method of the variation of parameters, 
first, because the necessity and sufficiency of the conditions which form the ground of 
that method and are the warrant of its success were to be shown ; secondly, because the 
connexion of the systems of ordinary differential equations which arise in the method of 
parameters with the successive partial differential equations forms an essential part of 
the demonstration. 
General Buie. 
The results of the foregoing inquiry may be collected into the following General 
Buie : — 
To find the number^ of possible integrals of a system of w differential equations of the 
first order connecting n-\-r variables ... x^+^y and to determine those integrals. 
Suppose P an integral of the given system. Determine from the given system 
dXi, dx^i-.-dx^ as functions of the other differentials. Substitute these values in the 
equation 
cW , , dP , 
T-dx,-}--!- dXn 
dx^ ^ ' dx^ 2 
dp 
dx. 
and equate to 0 the coefficients of the remaining differentials, 
of r partial differential equations of the form 
dP ^ 
dXn+i'^ " dx^ 
This will give a system 
dP 
dXn+; 
d^ 
dx^ 
r of the differential coefficients appearing each only in one equation and with coefficient 
unity. Eepresenting these equations in the symbolical form 
A,P=0, A^P^O,... A,P=0, 
deduce any equation or equations of the form 
(A,A,-A,A,)P=0. 
