PARTIAL DIFFERENTIAL EQUATIONS. 
159 
and 
In like manner 
Hence 
and 
t X x , dx r = e\^- dv - da 1 
d(u, v ) 
x lr = e 
d(x v x r ) 
0 /^ ^ \ 
X 2 , = 6 ~ J ’- , the multiplier 0 being the same. 
X„ = 0 
(rst) = 
c(u, V ) 
^(x rj x s ~) 
d(d, u, v ) 
3 (Xfj <^ 5 , 
Since this vanishes for all combinations of suffixes, 6 is a function of a, v, and if 
another function of them, w , is so chosen that 
3 w/du = 0 , 
we shall have £ X ra d(cc,., a?,) = 0 d(w, v) = c/(iy, y). 
r, s 
Linear Differential Equations. 
§ 10. If u = a is an integral of the linear partial differential equation 
X, 
+ X ; 
'»+! 
+ . . . + x.^» = x 
n +lj 
3®! 1 ‘ ti ’ 2 dx 2 1 1 3a?„ 
where X 1; X 2 . . . X * +1 are functions of x u . . . sc w+1 , then n satisfies the condition 
s‘ X,f^ = 0, 
r= 1 0X r 
and the complete differential du is a linear combination* of the determinants 
cZaq, c/tTg, doc^ ... , dccu^i j 
ll x lf X 2 , X 3 . . . X„ X « +1 | »• 
the coefficients in the combination being usually functions of x x , . . . as w+1 . 
If u = a is a common solution of the above equation and of 
Xi 
/ dx n+ i 
+ 
. +X„ 
■n 
/ ffi/t + l -y-' 
2, —-A. , <+ i, 
OX,, 
then, in like manner, du is a linear combination of the determinants 
* This is generally expressed by saying that “ u = a is an integral of the equations 
(JjOjO ’ (lX n +l ]] 
xi = = ‘ * = XXT • 
For the sake of the analogy with the work of §11, I prefer the phrase in the text, which expresses no 
more and no less than the one generally used. 
