278 
ON THE DIFFERENTIAL EQUATIONS 
[43 
or, reducing, 
1 1V7 fdA dA' \ 7 (dB dB' , 7 . 
V <iV + (s + W + ••■) &+ (a7 + AT + •••)*/ +••• = 0; 
K du dv 
whence separating the differentials and replacing A, A', ...; 5, B', by their values 
1 dV d du d dv 
V dx du‘ dx dv' dx~ '" ’ 
1 clV d d a d dv _ ^ 
V dy du'dy dv' dy ’ 
(in which — V, u, v ... ; x, y ... may be substituted for V, x, y ... ; u, v ...). 
§ 2. Let X, Y... be any functions of the variables x, y, ... and assume 
7 r , r du ,, du 
U = X — + Y + 
dx dy 
y _ v 4- V dv 1 
1 ~ X Tx +Y dy + '"’ 
U, V, ... being expressed in terms of u, v, .... Then 
dU^dV + _ y / d du d dv \ y / d du d dv 
du dv "' " \du'dx dv'dx "') \du' dy dv dy ^ 
+ I'dX du ^ dX dv \ fdY du dY dv 
du ' dx dv ' dx "'] \du ' dy dv ' dy 
i.e. 
^7 [dU dV \ / dV „dV \ _ fdX dY 
y l - IX— +F — + ...) +V( — + — + 
. du dv 
dx 
dy 
dx dy 
Also, whatever be the value of M, 
TT dMV Tr dMV 
ii T;- + F ir + 
and from these two properties, 
dMVU dMVV 
du dv 
v di¥V dJ/V 
„ fdMX dMY 
= Vi T + ^ + - 
§ 3. Consider the system of differential equations 
dx : dy : dz ... = X : Y : Z ... 
(where, for greater clearness, an additional letter z has been introduced). From these 
deduce the equivalent system 
we 
du : dv : dw ... = JJ : V : W ... .