uation itself. For instance, we may
include the case, sometimes arising when the equation to be solved is
obtained by transformation from another equation, in which F does not
contain either p or q. Then the equation has [oo]^2 solutions, each
consisting of an arbitrary point of the surface F = 0 and all the [oo]^2
planes passing through this point; it also has [oo]^2 solutions, each
consisting of a curve drawn on the surface F = 0 and all the tangent
planes of this curve, the whole consisting of [oo]^2 elements; finally,
it has also an isolated (or singular) solution consisting of the points
of the surface, each associated with the tangent plane of the surface
thereat, also [oo]^2 elements in all. Or again, a linear equation F = Pp
+ Qq - R = 0, wherein P, Q, R are functions of x, y, z only, has [oo]^2
solutions, each consisting of one of the curves defined by

dx/P = dy/Q = dz/R

taken with all the tangent planes of this curve; and the same equation
has [oo]^2 solutions, each consisting of the points of a surface
containing [oo]^1 of these curves and the tangent planes of this surface.
And for the case of n variables there is similarly the possibility of n
+ 1 kinds of solution of an equation F(x1, ... xn, z, p1, ... pn) = 0;
these can, however, by a simple contact transformation be reduced to one
kind, in which there is only one relation z' = [psi](x'1, ... x'n)
connecting the new variables x'1, ... x'n, z' (see under PFAFFIAN
EXPRESSIONS); just as in the case of the solution

z = [psi](y), x = [psi]1(y), [psi]'(y) = p[psi]'1(y) + q

of the equation Pp + Qq = R the transformation z' = z - px, x' = p, p' =
-x, y' = y, q' = q gives the solution

z' = [psi](y') + x'[psi]1(y'), p' = dz'/dx', q' = dz'/dy'

of the transformed equation. These explanations take no account of the
possibility of p and q being infinite; this can be dealt with by writing
p = -u/w, q = -v/w, and considering homogeneous equations in u, v, w,
with udx + vdy + wdz = 0 as the differential relation necessary for a
connectivity; in practice we use the ideas associated with such a
procedure more often without the appropriate notation.

Order of the ideas.

In utilizing these general notions we shall first consider the theory of
characteristic chains, initiated by Cauchy, which shows well the nature
of the relations implied by the given differential equation; the
alternative ways of carrying out the necessary integrations are
suggested