PARTIAL DIFFERENTIAL EQUATIONS. 
167 
and since dp, dq are absent we must have in each case 
Thus the equations 
h|' = 0, S A,^ = 0. 
r = 1 Op r = 1 Oq 
become 
dy — p l dx 1 + p z dx 2 , dz = q l dx 1 + qydx 2 
dip, 
du. 2 , 
du 3 , 
dip, 
dip 
du 2 
du-i 
dip 
dp ’ 
dp ’ 
dp’ 
dp 
dllry 
0M 2 
dip 
du 4 
dq’ 
dq’ 
dq’ 
dq 
These two equations, connecting du x , du 2 , du 3 , du 4 , taken with the system 
* 0 F 
• = 1 OUr 
4 0 F 
2 y a du r ~ 0 (a = 1, 2 or 1, 2, 3), 
show that if ?q, 
(a = 1, 2 or 1, 
solution, 
u. 2 , u 3 , u 4 satisfy by themselves no other relations than F a = 0 
2, 3) we must have, as a consequence of the equations of the 
w - ^ ^ q\ q\ v - / * 
r = l CU r Op r =l Oll r Oq 
If, then, there are two equations 
the four equations 
= 0, F 2 = 0, 
4 0F a 0« r 4 0F a du r 
2 w— w" = 0, 2 ,r— -x— = 0 (a = 1, 2) 
= I ou r op r= I me r oq 
must reduce to two only. This will be the ordinary case, and we see that if the 
forms of F : , F 2 , have been found by any means, the solution is completed without 
integration; the process corresponds to Charpit’s method of deducing all complete 
primitives from one, but it differs in that the functions F : , F 2 , are not arbitrary ; 
they must, in fact, be so chosen that the four equations last written shall reduce to 
two, and the conditions for this are clearly very complicated in general, though in 
particular cases available forms for F x , F 2 may be seen on inspection. 
In the more uncommon case, when there are three equations 
the six equations 
F 1 = 0, F 2 = 0, F 3 = 0, 
4 0 fq dip 4 0 F a du r 
r _ j du r cp ’ r = i du r dq 
0 (a = 1, 2, 3), 
must reduce to one only. 
These two cases are further discussed, from a somewhat different point of view, 
in §§ 21—23. 
It should not be forgotten that the form in which the new complete primitive has 
