452 
PEOPESSOK BOOLE ON SIMULTANEOUS DIEEEEENTIAL 
That there may be three integrals, it is here necessary that there should be but two 
partial differential equations in the completed system. Hence the equation 
(AA'— A'A)P=0 
must vanish identically. 
Developing this, we have the conditions 
AE+A'mi=0, Am2+A'T=0, A^— A|p=0. 
Now on performing the operations denoted by A and A', the last equation gives 
^2— »^l=0. 
Hence referring to the quadratic, we see that 
S^-4KT+4V=0 (I.) 
To this must be added the two other reduced conditions. 
AE+A'm=0, (11.) 
Am+A'T=0, (HI.) 
m representing one of the equal roots of the reduced quadratic. 
The first of the above conditions was given by Ampeee*. The others are probably 
new. Satisfied, they enable us to predict that the partial differential equation under 
consideration admits a complete primitive involving three constants, and a general 
primitive arising from the variation of those constants in subjection to any two arbitraiy 
conditions. 
3rd. We have supposed each linear partial differential equation employed in the pro- 
cesses of this paper to be of the form 
. dY dY . ^ 
and we have supposed each system of partial differential equations which arises, to be so 
reduced that each equation shall have some one of the partial differential coefficients of 
P entering into it alone and with a coefficient equal to unity. 
The first of these conditions is virtually sufficiently general, because any linear par- 
tial differential equation can be deprived of its second member. The advantage of the 
second condition is that each newly-formed equation will be really new, and not an 
algebraic combination of the old ones. 
But neither of these conditions is necessary. From two linear partial differential 
equations of the form 
A,P=H, A2P = K, 
in which H and K are functions of the independent variables, arises a new equation, 
(A1A2— A 2 A,)P=:AiK— A2H, ( 1 .) 
which will be satisfied by all the simultaneous mtegrals of the equations from wfiiich it 
is derived. 
It may be rigorously proved that, in applying this process, the generated system 
* Journal de I’Ecole Polytechnique, Cahier 18^ 
