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8. It is known that, if we are given any system of surfaces exponible 
by the partial differential equation of the first order 
F (2, Y, 3) DP; q) =0, 
we can, in general, determine the character of this system by finding the 
integral of the differential equation, either in the form of a complete pri- 
mitive, or a general primitive. 
Again, if we are given any system of surfaces exponible by the par- 
tial differential equation of the second order 
F (a, y, 8, p, 9, 7, 8, t) = 0, 
we may, in general, determine the character of this system by finding, 
if we can, the integral of the differential equation, either in the form of 
a complete primitive, or a general primitive. 
If the solution, in either of these two cases, be obtained in the form 
of a general primitive, the form of the arbitrary function or functions 
introduced is, in general, determined by supposing the surface to pass 
through a given curve or curves. The difficulty of applying this prin- 
ciple, in practice, is well known. 
Tf, on the other hand, the solution, in either case, be obtained in the 
form of a complete primitive, everything required by the solution is de- 
termined if, in the former case, to points upon the surface represented 
by the partial differential equation be given, and, in the latter case, five 
points. More generally, if the given non-linear partial differential 
equation be of the first order and include ~ independent variables, it is 
sufficient, for the determination of the function represented, in the form 
of a complete primitive, that we be given n systems of correspondent 
values of the variables. If the given non-linear partial differential 
equation be of the second order, and include » independent variables, it 
is sufficient for the complete determination of the function represented, 
that we be given —_— 8) systems of correspondent values of the vari- 
ables. 
9. Inow proceed to discuss certain examples with the view of show- 
ing the feasibility of obtaining solutions, in the form of complete primi- 
tives, of non-linear partial differential equations of the second order, and, 
in the following article, a general method for deriving such solutions 
will be indicated: the completion of the subject is reserved for a sup- 
plementary communication. 
