440 
PEOrESSOE BOOLE ON SIMULTANEOUS DIEEEEENTIAL 
we have 
An dx,... +A„i dx„=—dx„+,. 
A,, dx,... +A„, dx^= — dx„+r, 
which agrees with the system (IV.). 
Of the Determination of the Number of Integrals of a system of Differential 
Equations of the First Order and Degree. 
We still suppose the given system of differential equations to be expressed under the 
general form (I.), and reduced by Prop. I. to the equivalent partial differential system (II.). 
Now if for the expression of that system we introduce a series of symbols A„ Aj, ... A^ 
defined as follows, viz. 
( 1 -) 
the system will assume the form 
A.P=0, A,P=0,... A,P=0, 
. . ( 2 .) 
and we shall now establish the following proposition 
Peoposition 11.— If AjP=0, AjP=0 represent any two linear partial differential 
equations of the system (II.), then will the equation of which the symbolical expression is 
(AA-AA)P=0 (3.) 
also be a linear partial differential equation of the first order., and it will be satisfied by 
all the common integrals of the two equations from lohich it is formed. 
For, representing any one of the quantities ^i, x^, ...x„hy x, and any function of those 
quantities by X, A; consists of a series of terms of the form X Again, representing 
any one of the same series of quantities a:,, x^,... x^ by y, and any function of them by Y, 
Aj will consist of terms of the form Y Hence (AA 2 ““ AjAJP will consist of terms 
of the form 
(X;^ Y;^-y/x/)p. 
y ax ay ay ax J 
Effecting the differentiations, this term becomes 
or 
^ I XY— -Y — — -YX 
(lx dy "* dxdy dy dx 
X— ^_Y— — 
dx dy dy dx 
dxdy'' 
which involves only the first differential coefficients of P. Hence (3.) will be a linear 
partial differential equation of the first order. 
Hence also the ultimate form of (3.) will be the same as if A;, when operating on 
AjP, operated only on the coefficients A,j,- . . . A„^- involved in Aj, and rice versd. Thus the 
