117.J 
PARTIAL DIFFERENTIAL EQUATIONS 
203 
is an arbitrary function of the 2n — 1 particular solutions. Hence 
the general solution of the equation (7) is 
U 
where cf) n denotes any arbitrary function. 
This is a verification of the result of § 115 : and thus it follows 
that a first integral of the given differential equation is furnished 
hg any solution of the equation 
It is to be remarked;—first, that every integral of the original 
differential equation must satisfy this partial equation but that no 
integral, subsequent to the one initially taken, is completely deter 
mined by this equation:—secondly, that we may take any integral 
of the system 
dfl'i doc^ 
Vi ~ V2 
V‘in 
the subsidiary Pfaffian system, as an integral of the original 
differential equation, for this system is subsidiary to the complete 
solution of the partial differential equation ;—thirdly, that, even if 
V vanish (contrary to the initial hypothesis), yet, if not all the 
Pfaffians of order 2n — 2 vanish, the above partial differential 
equation (or the subsidiary system) is still valid for the determi 
nation of an element (p (§ 62) provided we retain the ratios of the 
vanishing quantities y*. 
118. We now pass to the case of a conditioned equation in 
( p =) 2m + q variables 
il = X 1 dxi —j— X2 dxa + H~ dxp — 0, 
a reduced form of which contains only m differential elements, say 
if — ffdfi + + F m df m , 
and we have to obtain the differential equations which are satisfied 
by the first of the integrals of il —■ 0 ; as in the general case before 
* The only essential difference between this case and the general case is that, in 
the present case, the fraction F r /F H admits of no simplification except the (possible) 
removal from the numerator and the denominator of a constant facto'r, while in 
the general case a variable factor thus disappears. See § 62.