655] 
A MEMOIR ON DIFFERENTIAL EQUATIONS. 
101 
19. The result would have been similar, if the adjoined equality had been = -™ 
(T a function of x, y, z, iv), but in reference to subsequent matter, I retain the 
equality = dt, and adjoin a second equality = (fi a function of x, y, z, w); we 
where H, X 
have then the integral t — r as before, and another integral 
are first expressed as functions of x, a, b, c, but after the integration a, b, c are 
replaced by their values as functions of (x, y, z, w), say this is the integral V — A; 
this, when the integrals a, b, c are known, is (like t — r) obtained by a quadrature. 
20. Attending only to the adjoined equality = dt, we can by means of the four 
integrals express each of the variables x, y, z, iv as a function of a, b, c, t — r; viz. 
these four equations, regarding therein t — r as a variable parameter, are in fact 
equivalent to the equations a = const., b = const., c = const., which connect the variables 
x, y, z, w with the integrals a, b, c regarded as constants. 
21. All that precedes is of course applicable to a system of n — 1 equations 
between n variables, the number of independent integrals being = n — 1. 
22. I take an example with the three variables x, y, z\ the differential equations 
being 
dx _ dy _ dz 
® (V ~ *) y{?-x) z(x-y)’ 
and therefore the partial differential equation 
The integrals are a = x + y + z, 6 = xyz; and it will be shown how either of these 
integrals being known, the system is reduced to a single equation between two 
variables, say x, y. 
First, a being known, = x + y + z as before, we have 
x(y-z)=x(x + 2y - a), y(z-x) = y (a - 2x - y), 
and the system is 
dx 
dy 
x (x + 2y — a) y (a — 2x — y) ’ 
which has the integral b = xy (a — x — y)\ observe that this is a solution of the partial 
differential equation 
x(x + 2y - a) -^+ y (a- 2x- y)^ 
For a putting its value we find b = xyz.