EQUATIONS OE THE EIEST OEDEK. 
441 
iiltimate form of (3.) will be 
(AAj-AjA,)§-+(A,A,j-AjA^)§^ +(AA,-A;A,)g=0. . . . (4.) 
Secondly, the equation (A^Aj— AjA,)P=0 will be satisfied by any common integral of 
AiP=0 and AjP=0. 
For let (p=c be a common integral of the latter equations. Then, identically, 
Ai®=0, Aj(p=0; 
therefore, since Aj and A,- involve only operations of differentiation together with alge- 
braic ones, 
A,A?)=0, A,.A^.^=0; 
AAj^-AjA^(p=0; 
whence (3.) is also identically satisfied. 
Pkoposition III. — If, by the successive application of Prop. II., and by permitted pro- 
cesses of algebraic elimination, we derive from the system of partial differential eguatiom 
A.P=0, A,P=0, A,P=0, 
into which the given system of differential eguations has been converted, a final system of 
partial differential equations which, ivhile including the above system, shall be such that 
the application of Prop. II. to any pair of the equations contained shall lead only to an 
identity, then the number of integrals of the given system of differential equations will 
be equal to the number of variables they contain, diminished by the number of partial 
differential equations of the above final system. 
The developed form of the system 
AiP=0, A2P=0, ... A,P=0 (1.) 
is the following ; — 
, If . ^ 
dxn-^i ”^dx„ ’ 
dP 
I A ^ I A ff Q 
I A t A ^^P n 
dXn+ff ^’'dx^ 
( 2 .) 
Comparing these with (4.), Prop. II., which is the developed form of the equation 
(AjA^— AjA,)P=0, we see that the latter equation is necessarily algebraically independent 
of the above system ; for no equation derived from that system by algebraic processes 
dV rfP 
could be free, as (4.) is, from all the differential coefficients — ••• 
Again, as (A.Aj— AyAi)P=0 is satisfied by all the common integrals of AiP=0 and 
A^P=0, it follows that the system of r-f 1 equations, 
AiP=0, ... A,P=0, (AA,-‘A,A,)P=0, (3.) 
will be satisfied by all the common integrals of the system (1.). To this system of r+1 
