[655 
655] A MEMOIR ON DIFFERENTIAL EQUATIONS. 105 
expression 
From the foregoing values of MX, MY, MZ, we deduce 
MU M V ^ (u, fy d (y, a, b) 
d {x, y, z) ’ d (x, y, z)' 
But forming the values of du, dv, da, db, we have an equation, determinant = 0, which 
equation may be written 
dn d X a ’ b \ dv d M l ’ u \ + d a d X u ' v \ db d X V ’ tt l-0; 
d (x, y, z) d (x, y, z) d (x, y, z) d (x, y, z) 
or, writing herein da = 0, this is 
du d X a ' b \ dv d M a ’ b \ db d X' V ’ a \-0, 
d (x, y, z) d (x, y, z) d (x, y, z) 
viz. this is 
M ( Vdu Udv) = db d {V ’ a) 
a (x, y, z) 
or say 
= X + y + z, 
where, on the right-hand side, everything must be expressed in terms of u, v, a. It 
thus appears that on expressing the final equation as a relation Vdu — Udv = 0 between 
the variables u and v, the multiplier hereof is M 4- ( u >_ v >_ a ) _ If u, v —x, y, this agrees 
d(x, y, z) 
with a foregoing result. 
e b = xyz, 
29. The theory is precisely the same for any number of variables. Thus, if there 
are four variables x, y, z, w, we have 
MX MY M r/ MW — d ( a ’ c ^ d(a, b, c) d (a, b, c) d(a, b, c) 
’ ’ ’ d{y, z, wy d(z, w, x)’ d(w, X, yY d(x, y, z y 
and, we have between the functions on the right-hand an identical relation, in virtue 
of which 
d(MX) i d(MY) ( d(MZ) ^ d(MW)_ Q _ 
dx ' dy dz div 
by taking 
then, supposing that a value of M is known, and also any two integrals a, b, and 
that by means of these the equation to be finally integrated is expressed as a relation 
Vdu — JJdv = 0 between any two variables u and v, the multiplier of this is 
_ ^ . d (u, v, a, b) 
d (oc, y, z, w) ’ 
functions 
where U, V and this multiplier are to be expressed in terms of u, v, a, b. 
* in this 
The general result is that, given a value of the multiplier, and also all but one 
of the integrals, the final integral is expressible by a quadrature. 
c. x. 14 
14