442 PEOFESSOE BOOLE ON SBIULTANEOUS DIPEEEENTIAL 
equations we can also give a form analogous to the developed form of the system (2.). 
dP dP 
It will be noticed that the differential coefficients — - ... appear there, each in 
only one equation, and each with the coefficient unity. Now let the last equation of 
(3.) in its developed form be dhided by the coefficient of—, and let also the value of 
(IXji 
dP 
— which it gives be substituted in the other equations of (3.) ; then we shall have in 
(xXfi 
the whole a system of w+1 equations possessing the same general character as the 
system (2.). To this new system the same procedure may be applied, viz. the genesis 
of a new equation by means of Prop. II., and the transference of another differential 
dP 
coefficient ^ to the list of those which form the respective first terms of the equations 
(tXn—\ 
of the system. We will suppose this procedure to have been repeated until a system 
composed of m partial differential equations sw/i that the further application of Prop. II. 
leads only to identities has been formed. If n-\-r—m—p., that system will be of the 
form 
dp 
[-H 
dP 
■■ +H,. 
dXp q- 1 
dP 
dp 
1 JJ 
dXp 
dXp-\-<2, 
dP 
T-r dP 
-TT dP 
f! 7’ 
'• '^^P'^'dxp 
(^•) 
And if, in analogy with former notation, we write 
4;:+h4+h4.,.+H,,^=a„ 
it will take the symbolical form 
, _ _ A,P=0, A,P=0,... A,„P=0; 
but it will differ from all former systems of equafions in that all the conditions repre- 
sented by 
(a,a-a,a,)I’=o (6.) 
will be identically satisfied, — satisfied, in consequence not of any ascertained peculiarity 
of the integral P, but of the constitution of the system of symbols Ai, A 2 , . . . A,„. 
The course of argument has shown that the common integrals of the system (4.) wall 
be identical with those of the parent system (2.). Now we shall show that the exist- 
ence of the condition (5.) renders the integration theoretically possible ; that the system 
of^ ordinary differential equations into which, by Prop. II., the final system of partial 
differential equations (4.) is resolvable admits of exactly integrals. A.S, ])=n-\-r — ni. 
this is to say 'that the "number of integrals is equal to the number of original variables 
diminished by the number of final partial differential equations. 
