384 
It is easy to see how the following non-linear partial differential 
equations are but modifications of the simple examples treated, namely, 
d Sal ial adie sls Soy ak 
Peg do * do * To we 
dz dy ds 
| |B cee wil 
Bhs fsa al pier 4 de * Bo * Teh" 
2 q ac dy de 
TRONS ee ee kw 
J de dy dz 
the solutions being, respectively, 
1l/ « y 
log s =3( era 
1/2 ¥# | 
2 = — 
: z( hie ee) ( 
1/gm™ ym | 
m+] — _ 
A = a et, 
and 
+ 
wee “(E65 tnt] 
(3). If it were proposed to integrate the non-linear partial diffe- 
rential equation of the second order, in two independent variables, 
ti, 71 1 
rt @s! By 
die dy 
we would assume, by analogy, the solution to be 
xe? y wy Lf Yy 
2a. Hla) Big. 8 
and it is instantly evident that this is a complete primitive of the given 
equation, since it satisfies it, and exhibits five arbitrary constants. 
1, 
i sp G5 
