385 
Similarly, the solution, in the form of a complete primitive, of the 
non-linear partial differential equation of the second order, in three inde- 
pendent variables, 
1 1 ug 
dot to* Fo” 
da? dy? ae 
is 
— EWS 3 ys sa wy y 
w= ae ai Bt ® 
* 987 2(1 - a+) Ss 
since, as before, it'satisfies the given partial differential what and ex- 
hibits nine arbitrary constants. 
It is readily seen that if the former of the two partial differential 
equations now treated, had been 
dx? dudy dy? 
we should have, for a ee primitive, the obvious modification 
cs ve Mes play 
es +8 + 1 2ia 8B) 
(4). It follows readily from the forms of solution arrived at in the 
examples just discussed, that the complete primitive of any non-linear 
partial differential equation of the type 
FS (2; 9) = 9, 
+= sists 
is the linear expression 
s=ant+ byte, 
where @ and d are connected by the equation 
F (@, 6) = 0. 
Thus the complete primitive of the partial differential equation 
1 
al 2 =k 
5 gee. 
is 
s= z+ Z see (2B 
a™ re! xs a)” 
B. I, A. PROO.—VOL. VI. 3T 
