and if we write the single letters A, B, C, D, E, F, G for al (u, v, w\, al (u, v, w) 2 , 
al {u, v, w) 3 , al (u, v, w) 4 , al (u, v, w) 5 , al (u, v, w) 3i al («, v, w) 7 respectively, each of the 
capital letters thus denoting a function of (u, v, w), the expressions of these functions 
in terms of (x, y, z) are 
A = *Ja — x.b — x.c — x, (seven equations). 
Similarly, instead of the 21 functions al(u, v, w) l2 , ..., al (u, v, w\ 7 writing AB, ...,FG, 
each of these binary symbols denoting in like manner a function of (u, v, iv), the 
definition of AB is 
AB = A VB-BVA, 
where 
d d d 
^ du + dv + dw ’ 
we have 
b — c.c — a.a — b.^y = -——-(6 — c)du + ^——-(c —a)efo + C 2/ ,c 12 
yjL X — y.X — Z X — y.X — Z i 
dy a — z.a — x n x7 b — z.b — x. c — z.c — x, 
= (b -c)du-\ (c -a)dv-\ (a - b) dw, 
y Y y — z.y—x y-z.y-x y — z.y — x 
hence 
dz a — x .a — y b — x.b — y. . , c — x.c—y, 7 , 7 
= (b — c) du H (c - a) dv-1- (a -b)dw; 
V z — x.z — y z — x.z — y z — x.z — y 
b — c.c — a.a — b „ a- y.a—z 
V x = 
/7 \, b-y.b — z c-y.c-z. 
(6 - c) + -——- (c - a) + —A ( a _ M 
that is, 
and similarly 
Hence from the equation 
we have 
x — y.x — z X — y.X — Z 
b — c .c — a. a — b 
x — y.x — z 
V x = — 
y — x.y — z z — x.z — y 
A = *Ja — x.a — y.a — z 
that is, 
and similarly 
VA=-±A f-i— V«+— Vi/ + — vA 
\a —« a — y a — z J 
y — z.z — x.x — y [a — x a ~y a — z ) 
WB = 
consequently 
y — z.z — x.x — y 
+ + 1; 
(6 — a? 6—i/ 6 — z J 
= j- (^ ~ 6) AB ((y-z) \JX + (z-x)s/Y + (x - y) ) 
y — z.z — x.x — y\a — x.b — x a — y .b — y a — z.b — z) ’ 
C. X. 
55 
1 
rdj.