PARTIAL DIFFERENTIAL EQUATIONS. 
173 
in the third the same expression with 0, xjj interchanged. The array is thus practi¬ 
cally unchanged by interchanging the sets x and u, as should be the case. 
§ 24. This transformation may be accomplished by taking the equations 
00 dui 
from which may be deduced 
. 00 dui „ . 
v —L. —- — o — y 
i cb\ i dui dx 2 ’ 
or 
d 00 duA d / 00 du 
dXy \i dui dxj dx 2 \7 dui dx v 
^ v 0 2 0 dui duj c~(j) duj 0-0 dm 
i 7 duidttj dx 3 dx 1 i dui 0q dx 3 ; civ.Ay dx 3 
0-0 dui duj . 0-0 du[ , 0-0 dui 
+ 
0-0 
i auioz dx„ 
0-0 
dui 
dx 3 
du; 
= 22 3331=13 , 2 <?£ , 2 , 2 „ 
i j duidiCjdxydx 3 * duAx 3 da\ * du i dy*'^dx 1 t duidz'd' 2 dx l 
Now p v q lf p 2 , q. z are given by the relations 
tty , , 3</> _ n o 
^ + Ni + ?i a: — °> &c -’ 
ox 
03 
and hence this equation may be written 
2 r *, 8 (*■*•!! ' 
cfo, y, z) 
d ;F i d(.r 3 , y, z) 
(ii); 
in this (f >, 0 may be interchanged so as to give another equation. 
Now, suppose 6 = aq, y = a 2 to be two of the four equations connecting 
iq, m 3 , u 3 , with aq, x 3 , which yield a new complete primitive, and that y, z have 
been eliminated from 6, y by means of the equations 0 = 0, 0 = 0, then the deriva¬ 
tives yy, yp, &c., are given by the following relations :— 
00 dui _ 
i dui dbq ’ 
00 dMi _ 
7 3 «I ^ “ ’ 
. dd dUi 
i oui dx. 
vJL 
0..q 
= 0 , 
V 0 % dip , 0 y 
7 diii dx\ dx 1 
and similarly for the derivatives with respect to x 2 . 
Substituting the values hence found for these derivatives in the equation (11), we 
have an equation linear in the Jacobians of the form 
9(0, %) 
d(Xi, uj ) 
(» = 1.2; 7 = 1,2, 3, 4), 
