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< be a solution of the differential equation, then
dz = dxd[psi]/dx + dyd[psi]/dy; thus if we denote the equation by F(x,
y, z, p, q,) = 0, and prescribe the condition dz = pdx + qdy for every
solution, any solution such as z = [psi](x, y) will necessarily be
associated with the equations p = dz/dx, q = dz/dy, and z will satisfy
the equation in its original form. We have previously seen (under
_Pfaffian Expressions_) that if five variables x, y, z, p, q, otherwise
independent, be subject to dz - pdx - qdy = 0, they must in fact be
subject to at least three mutual relations. If we associate with a point
(x, y, z) the plane
Z - z = p(X - x) + q(Y - y)
passing through it, where X, Y, Z are current co-ordinates, and call
this association a surface-element; and if two consecutive elements of
which the point(x + dx, y + dy, z + dz) of one lies on the plane of the
other, for which, that is, the condition dz = pdx + qdy is satisfied, be
said to be _connected,_ and an infinity of connected elements following
one another continuously be called a _connectivity_, then our statement
is that a connectivity consists of not more than [oo]^2 elements, the
whole number of elements (x, y, z, p, q) that are possible being called
[oo]^5. The solution of an equation F(x, y, z, dz/dx, dz/dy) = 0 is then
to be understood to mean finding in all possible ways, from the [oo]^4
elements (x, y, z, p, q) which satisfy F(x, y, z, p, q) = 0 a set of
[oo]^2 elements forming a connectivity; or, more analytically, finding in
all possible ways two relations G = 0, H = 0 connecting x, y, z, p, q
and independent of F = 0, so that the three relations together may
involve
dz = pdx + qdy.
Such a set of three relations may, for example, be of the form z =
[psi](x, y), p = d[psi]/dx, q = d[psi]/dy; but it may also, as another
case, involve two relations z = [psi](y), x = [psi]1(y) connecting x, y,
z, the third relation being
[psi]'(y) = p[psi]'1(y) + q,
the connectivity consisting in that case, geometrically, of a curve in
space taken with [oo]^1 of its tangent planes; or, finally, a
connectivity is constituted by a fixed point and all the planes passing
through that point. This generalized view of the meaning of a solution
of F = 0 is of advantage, moreover, in view of anomalies otherwise
arising from special forms of the eq]]>
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