98
A MEMOIR ON DIFFERENTIAL EQUATIONS.
[655
The General Differential System. Art. Nos. 9 to 22.
9. Taking the set of variables to be (x, y, z, w), the system is
dx _dy _ dz dw
X = Y = X == W’
and we associate with this the linear partial differential equation
x c M + ydO + z ( W + w dd = 0.
dx dy dz dw
10. It is tolerably evident that the differential equations establish between
x, y, z, w a threefold relation depending upon three arbitrary constants; in fact,
regarding (x, y, z, w) as the coordinates of a point in four-dimensional space, and
starting from any given point, the differential equations determine the ratios of the
increments dx, dy, dz, dw, that is, the direction of passage to a consecutive point;
and then again taking for x, y, z, w the coordinates of this point, the same equations
give the direction of passage to the next consecutive point, and so on. The locus
of the point is therefore a curve, or we have between the coordinates a threefold
relation, and (the initial point being arbitrary) we have a curve of the system
through each point of the four-dimensional space, viz. the relation must involve three
arbitrary constants. But this being so, the constants will be expressible as functions
of the coordinates, viz. the threefold relation involving the three constants will be
expressible in the form a = const., b = const., c = const., where a, b, c denote respectively
functions of the coordinates (x, y, z, w).
11. Supposing that one of the relations is a = const., it is clear that the increment
da,
da ,
-j- dx 4-
dx
da
dy
dy +
da
dz
dz +
da
dw
dw,
must become = 0, on substituting therein for dx, dy, dz, dw, the values X, Y, Z, W
to which by virtue of the differential equations they are proportional, viz. that we
must have identically
Xp + Y da + Z~-+ = 0.
dx dy dz dw
Conversely, when this is so, we have da = 0, by virtue of the differential equation.
We say that a is a solution of the partial differential equation, and an integral
of the differential equations, viz. any solution of the partial differential equation is
an integral of the differential equations, and any integral of the differential equations
is a solution of the partial differential equation, or, this being so, we may in general
without risk of ambiguity, say simply a is an integral*; similarly b and c are
integrals, and, by what precedes, there are three integrals a, b, c.
* Viz. we use indifferently, in regard to the differential equations and to the partial differential equation,
the term integral, which is appropriate to the differential equations ; the appropriate term in regard to the
partial differential equation would be solution.
