102 
A MEMOIR ON DIFFERENTIAL EQUATIONS. 
[655 
Secondly, b being known, = xyz as before, we have 
and the system is 
(y-z) = xy-^, y(z-x) = ~-xy, 
dx dy 
b ~b ’ 
xy xy 
J y X * 
which has the integral a = x + y + —; observe that this is a solution of the partial 
xy 
differential equation 
b\ dd (b \ dd 
+ l=-»)ar a • 
^ y) dx ' \x 
For b putting its value, we find a = x +y + z. 
The Multiplier. Art. Nos. 23 to 29. 
23. First, if there are only two variables (x, y), the system consists of the 
single equation 
dx dy 
X = T’ : 
which may be written 
Ydx — Xdy = 0. 
Hence, if a be an integral, we have 
da 7 da 7 
dx + 7 dy = 0 : 
dx dy 
the two will agree if there exists a function M such that 
~=MY, ^ - MX, 
dx dy 
and thence, in virtue of the identity 
we find 
or, as this may also be written, 
dM 
d da _ d da 
dy dx dx dy ’ 
dMX dMY 
dx + dy ’ 
v dM dM ,, (dX dY\ A 
X -d, + r dy +M U + -dy) = 0 ’ 
as the condition to determine the multiplier M. Supposing M known, we have 
M(Ydx — Xdy) = da, or say a = JM (Ydx — Xdy), viz. the integral a is determined by a 
quadrature.