446 
PEOEESSOE BOOLE ON SIMTILTANEOES BIEEEEENTIAL 
j=l and P=w„ we have 
(AfAi— A,A>,=0. 
But AiW,=0; therefore, by the above, 
Now AjW^ is expressible at most as a function of u ^, ... Wp+i ... 
formation converts, as has been seen, Aj into ^ — . Thus we have 
But this trans- 
^p+i 
d 
dx 
/?+l 
■Ai^i,=0, 
Ap, the 
so that A{tJb, is free from Xp^^. Thus the system (11.) is free from Xp^^. 
Lastly, since by the above transformation Ag ... A^ are converted into A^ 
system of conditions (AjA^- — A^Ai)P=0 is converted into (A'^A) — A)A-)P=0. 
It is thus seen that the system of^ partial differential equations 
AiP=0, A2P=0, ... A^P=0, 
containiag independent variables x^, X 2 , ...Xp+m-, and satisfying the conditions 
(aa-aaF=o. 
is convertible into a system ofp— 1 partial differential equations, 
a;p=o, a'3P=o, a;p=o, 
containing independent variables ... Up^ Xp +2 and satisfying the con- 
dition 
(aA-aa)p=0- 
And as this system possesses the same character as that upon which the previous trans- 
formation depended, it will admit of transformation into a system of ^—2 partial differ- 
ential equations containing p-\-m — 2 independent variables; and so on until we arrive 
at a single final partial differential equation containing ^-}-l independent variables, and 
having therefore p distinct integrals, which will be the common integrals of the primary 
system of partial differential equations as well as of the system of ordinary differential 
equations to which they correspond. 
Co7\ The property of the coefficients AgW,, &c. of the system (11.), of being free 
from the variable ^^+ 1 , enables us, by properly determining the integrals of the par- 
tial differential equation A,P=0, to reduce the system to a form of great simplicity. 
Let AiUj be any one of those coefficients. Its developed form is 
(^,+h4-+h.4)“' 
Now as this expression will, after the performance of the differentiations, be free from 
and as the differentiations are none of them with respect to Xp+i, we can give to 
Xp^■^ in it any particular value before differentiation without affecting the final result. 
Let us then suppose that in ... H^;, and in Wy, Xp^^ is made equal to 0. Now it is 
possible so to determine the integrals Uj as functions of the variables x ^, ... Xp+^, that 
