444 
PEOPESSOE BOOLE ON SIMULTANEOUS DIEEEEENTIAL 
2ndly. We proceed to show that the system of m linear partial differential equations 
(7.), represented under the form 
A,P=0, ... A„P=0, 
and satisfying identically the system of conditions represented by 
(AiA,— A,A;)P=0, 
will admit of^ integrals expressing distinct values of P ; and the system of ordinary dif- 
ferential equations (6.) corresponding to the above system of partial differential equations 
will be expressible as a system of exact differential equations, and will by integration 
give the above systems of integrals. 
Beginning with the first partial differential equation of the system (4.), and formmg 
the corresponding Lagrangian system of ordinary differential equations 
- dx^ 
dxp 
dXp^m — 0, 
we see that the integrals of this system will be of the form 
u. 
—c, 
.u„ 
—c, 
pi 
^p+2 ^p+2 • • • ^p + m~~^p+m1 
Uy...Up being functions of all the variables . . x^^^^ among which, by virtue of the 
integrals of the second line, Xp ^^^ . . . Xp^^ may be regarded as constant. The general 
integral will be 
. . . Wp, Xp ^^^ . . . Xp+^ — 0 , 
the form of F being perfectly arbitrary. 
Now the general form of any equation of the system (4.) is 
.jjd? 
dxp+i ' dxi 
(9.) 
Let us transform this by assuming 
Ul t . . Up^ ^p + i • • . ^p+m 
as independent variables. Then referring the right-hand members of the following 
equations to the new, the left-hand to the old system of variables, we have 
dV 
du^ 
• • + 
dV 
dup 
doCp+i 
dx.pj^i 
* (/Mj 
dXp+i • 
dup 
dXp^i 
d? 
dV 
du. 
dV 
dup 
dx^ 
^ . 
dUy 
dx^ 
dup 
dx^ 
dV 
dV 
du. 
dV 
duo 
dxp 
duy^ 
dxp 
dUp 
dXp 
