EQUATIONS OF THE FIEST OEDEE. 
439 
shows that by an exactly similar rule any system of n partial differential equations of 
the first order, the terms of which consist of the differential coefficients of P multiplied 
by functions of the independent variables x^-,... x^^r-, may be converted into an equi- 
valent system of r common difierential equations of the first order. 
For the proposed system of partial difierential equations is by algebraic reduction 
expressible in the form 
— —A 
dV 
dXa 
: Aji 
^=A - 
dxn dx, 
dXfi^i 
dV 
dXn + \ 
rfP 
-A, 
■A„ 
n + l 
d^ 
..-f-Air 
JP I 
dXn+2 
dXn^r^ 
dV 
. .+A2;. 
dV 
dXyi J^2 
dXn+r^ 
<ZP 
dV 
dXn+2 
dXn+r 
(III.) 
If the values of ^5 ^5 • • • ^ in this system be substituted in the previous general 
equation, and the coefficients of the difierential coefficients 1 5 ^ 5 ‘"-i — in the 
result be separately equated to 0, we shall have 
fZ^„+,+Ai, fZ^i+Aji ^7.^2 . . .+A„i fZx„=0,' 
fZa’n+a + A i2 dx-^ + A 22 dx ^ . . . +A„2 <Za‘„ — 0,1 . 
Ja’„+^+A„ tZ^i+Aa, fZ^2 •• <Z.r„=0. 
These equations qiay in like manner be formed by inspection from the columns of 
rfP 
the second member of (HI.), by writing for -1 in any column dx^+i, adding to this 
(ZoTi, dx^t . . . dXn multiplied by the descending coefficients of the column, and equating the 
final sum to 0. The rule for the one case differs from that for the other only in that 
differentials take the place of differential coefficients. 
As an objection may be felt to the legitimacy of that step of the above process in 
<ZP 
being eliminated, the coefficients of the 
JP «/P 
which, the differential coefficients -7-’ 
remaining ones are separately equated to 0, I wall point out another mode of procedure 
which leads to the same result, and which is founded upon Lagrai^ge’s method of 
solution. Let the equations of the system (HI.) be added together after having been 
multiplied respectively by X,, >^2, ... X„, which are to be regarded as indeterminate func- 
tions of the variables x^^x^^... .r„+^. The result will be a linear partial differential equa- 
tion of the first order, of which the Lagrangian auxiliary system of ordinary differential 
equations will be 
dx^ dx^ 
dXn 
— dXn 
— dx. 
Tt-f-r 
Xj Ag A^ Aj 2 Aj^ Ani A;j Aj;. Aj Atit* A;j 
Hence, eliminating Xj, Xj, . . . or more strictly speaking the ratios 
A„’ 
An-I 
A;, ’ 
3 0 2 
