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Examples. 
(1). Let it be proposed to determine the solution, in the form of a 
complete primitive, of the non-linear partial differential equation 
Tee 
eamee fl 1 
DL Ag a) 
The auxiliary system, adopting the method of Lagrange and Charpit, 
gives 
dp = 0, 
since 
dq dq 
de agp 
Hence, ¢ being any arbitrary constant, 
pr=e , and q= 3 
and substituting these values in the general equation 
dz = pda + qdy 
we get, writing + for c, 
y 
s=< +2 +f (2) 
This then is @ complete primitive of the given equation, and it is at once 
seen to satisfy it. But, moreover, if we examine the other equations of 
the auxiliary system in this case, namely, 
(p - 1) da = dy = (2) di, 
2p 
we see that no other complete primitive is deducible. In other words, 
equation (2) is the complete primitive of (1). 
Tf B be made an arbitrary function of a, we get as the general pri- 
mary of (1) 
a iliac det 
oe a eae ee) | 
(2). If it be now proposed to integrate the higher differential equa- 
oe ge 
