539 
1907-8.] Algebra after Hamilton, or Mnltenions, § 7. 
Let u v u 2 .... u be k multenions satisfying the conditions 
Sm 1 |m 1 = Sw 2 [m 2 .... =1, = Si/Jttg = .... =0 . . (34) 
It is easy to prove that they must be independent, and are capable of 
being obtained from the multits by circular variation. The n fictits are 
one particular case of u v u 2 , .... u k and the 2 n multits are another 
particular case. 
Let all the multenions considered belong to the complex u v u 2 , ... . u k . 
If a = HxL and q = ^yu, x and y may be called a co-ordinate of a 
and a co-ordinate of q respectively. Thus Sa|/3 is the same function of 
the co-ordinates of a and /3 that Sp|g is of the co-ordinates of p, q. Hence 
we define the conjugate <fi' of a multilinity cp by 
Sr|</>'s = Ss|<£r (35) 
for all multenion values of r, s. The reader is advised to omit the details 
from this point to eq. (54) till an occasion when he finds he requires them 
for other applications, f may be paralleled by £ according to the equations 
L(f,J) = SL( l ,0,L(&« = SL (u,u) .... (36) 
whence parallel with (22) § 6 we have 
q = mq (37) 
Hence, omitting the last member of (11), all the equations (1) to (18) 
may be read as multilinity equations by merely substituting 
p, q,t; r, s; £ 
for a, f3, y; p, o- ; f . 
Let 
q = au Y + bu 2 + ... . 
+ lu k 
q^ = d-^u-^ -f- b-^u 2 + • ■ • 
. +l Y u k 
9k-i ■= d k _ x u x + + . 
. . . + l k -\U k j 
(38) 
be k given multenions. Then define the “combinatorial scalar” C s q {k) = 
0 8 {qq 1 .... q k _P) of k multenions q, q x ... . q k _ x ; and the “ combinatorial 
multenion ” C m q (k ~ 1} = C m (q 1 q 2 .... q k _-P) of k — 1 multenions by the equations 
a 
b 
. . i 
C s q w = 
h •• 
.. z, 
. (39) 
ajc-i 
bjc-i • • 
• • 4-i 
P 
& 
1 
G s qq {k ~ 1] = 
-i) 
• (40) 
. . . L are the 
minors 
corresponding to 
the first 
row of 
( 39 ) 
