[655 
655] 
A MEMOIR ON DIFFERENTIAL EQUATIONS. 
103 
partial 
of the 
we have 
led by a 
24. In the case of three variables (x, y, z), the system is 
dx _ dy dz 
X^T = ~Z’ 
or, writing these in the form 
Ydz — Zdy = 0, Zdx — Xdz = 0, Xdy — Ydx = 0, 
the course which immediately suggests itself is to seek for factors L, M, X, such 
that, a being an integral, we* may have 
L (Ydz — Zdy) 4- M {Zdx — Xdz) + N {Xdy — Ydx) = da, 
but this does not lead to any result. The course taken by Jacobi is quite a different 
one: he, in fact, determines a multiplier M connected with two integrals a, b. 
25. Supposing that a, b are independent integrals, we have 
y da yt da „ du „ 
dx dy dz 
X~+Y~ + Zj b =0] 
dx dy dz 
and determining from . these equations the ratio of the quantities X, Y, Z, we may, 
it is clear, write 
MX, MY, MZ= d /. a, - b \, d ^r h l- 
d {y, z) d {z, x) d {x, y) 
It may be shown that we have identically 
d d{a,b)^d d{a, b) d d{a, b) ^ 
dx d {y, z) dy d {z, x) dz d {x, y) ~ ’ 
and we thence deduce 
or, what is the same thing, 
d {MX) d {MY) d{MZ) 
dx + dy dz 
v dM v dM „dM . fdX. dY dZ\ 
X cfc + + Z dz +11 (ST + dy + s) -°- 
as the condition for determining the multiplier M. 
26. The use is as follows: supposing that M is known, and supposing also that 
one integral a of the system is known, we can then by a quadrature determine the 
other integral b. Thus, supposing that we know the integral a, —a {x, y, z), we can 
by means of this integral express z in terms of x, y, a; and hence we may regard 
the unknown integral b as expressed in the like form, b = b {x, y, a). The original 
db db db 
dz 
db da 
values of 
dx’ dy’ become on this supposition 
db db da db db da 
dx da dx ’ dy da dy ’ da dz’