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and y, and eliminate the arbitrary constants between the given equation 
and 
(2) = () =» (2-9) -a()-* 
Thus we should obtain a partial differential equation of the second order, 
in general non-linear, of the form (adopting the ordinary notation), 
F (x, Y, % P,% 7; 8, ¢) = 0, (IV) 
of which (III) may be said to be a complete primitive. 
6. It does not follow, however, (as is known), that, in general, from 
a given finite equation exhibiting two arbitrary functions, we can pass 
to a partial differential equation of the second order. For if, w and v 
being any two functions of known form in @, y, s, we were given any 
equation of the type 
S(@, Y, & QU, yr) = 0, 
differentiating this equation twice successively with respect to x and y 
we only get six equations, which, ordinarily, are no¢ sufficient to enable 
us to eliminate the siz quantities 
$959 0, vp". 
It is true indeed that, in the case of linear partial differential equations 
of the second order, which admit of treatment by symbolic methods, we do, 
in most cases, obtain the solutions, in the form of general primitives, by 
direct procedures, and exhibiting two arbitrary functions. But the fact, 
which has just been alluded to, would appear to show that in the case 
of non-linear partial differential equations of the second and higher or- 
ders, which do not admit in general of treatment by symbolic methods, 
the species of solution which we should seek to obtain, should be, not 
the general primitive exhibiting arbitrary functions, but the complete 
primitive exhibiting arbitrary constants; and the mode of integration 
which we should seek to perfect, should be that by which the complete 
primitives of such partial differential equations are sought, and not their 
general primitives. 
7. Moreover, we may differentiate the equation. 
Sy, z, a, B, ¥, 9, e)=0 
with respect to the arbitrary constants, and between the five equations 
thus found, namely, 
ie Ce eee ig Oh vg Lh 
Bat MUR ipo Ba deg’ 
and the given equation, eliminate a, P, y, 6, «, thus obtaining a result- 
ing equation of the form 
F, (@, y, z)=0. (V) 
