438 PEOFESSOE BOOLE ON SIMXJLTAJSEOUS DIFFEEENTIAL 
tions may, by algebraic solution with respect to dx,, dx ^,'. . . dx^,, be reduced to the form 
dXy'=^A.iidXn+\~\~ Ai2^?.r„+2 • . • -\~A.ij.dXn+r-) 
dx3=A2idx„+^ +A 22dx, 
n + 2 
A.2ydx, 
»+»■> 
(I-) 
dX^ — A„j6ZtJ7„^j -j“ A„2^?^n+2 • • • “h -^nr^^n+rl 
the coefficients An, A 12 , &c. being functions of the variables 
Let P=c be an integral of the system. Then 
^-^dxAx 
dx. 
dXn+^=0. 
n+r 
Substituting in this equation the values of dXy, dx^, .. .dx„ given by (I.), we have 
dP 
( 
+ (dx.. 
dXn+i 
dP 
A ^ , A ^ 
" fifej ' dx^ 
, dP . dP . 
• • • + A„, w„+i 
^dXn 
dx, 
dx. 
n+2 
+ 
dP 
dXrt 
• • • +^«>- dx^ ^^n+r— 0- 
As the differentials «?.r„+i, dx^^^:, • • • d,Xn+r independent, we have, on equating 
their coefficients separately to 0, 
dP 
dXn + 1 
dP 
dXn+2 
, . . ^?P ,xdP 
■r^2i •• • — G, 
^dxi 
''^^dx^ 
■A, 2 ~ “hAj 
1-2 
'■dx,. 
dXn 
d^ 
dXn 
'• +A„2 0, 
, , dp ^ 
dx„ + r'^‘^^’-dx, • '^■^"’'dXn — ^^ 
(IL) 
a system of r linear partial differential equations, the common integrals of which will be 
the integrals of the system (I.). We say ‘ the common integrals of which,’ because in 
fact these equations express the conditions to all of which P must be subject in order 
that P=c may be an integral of the system (I.). 
The formal connexion between the systems (I.) and (II.) deserves to be carefully 
noticed. The several partial differential equations of the system (II.) may be formed by 
inspection from the columns in the right-hand member of the system (I.) by the follow- 
ing rule. For the differential dx„+i in any column write the differential coefficient 
dP . . dP dP dP . . . 
-j - — 5 to this add the series of differential coefficients ...x~ multiplied in succes- 
CLOC^ ClOC(^ (tOCf^ 
sion by the descending coefficients of the column, and equate the final result to 0. 
The symmetrical form of the equation 
. dP 
dP ^ dP ^ 
dx^ + dx,f^^ 
dx, 
n+r 
