1907-8.] Algebra after Hamilton, or Multenions, § 11. 581 
The connection between these sums and the coefficients is well known 
and is as follows. If the identity is 
gk _ h'gic-i + ft'qk - 2 0 
then kSq - ti = 0 
kSq*-h'.kSq + 2ti’ = 0 
kSq* - h'.kSq 2 + h".kSq - 3 h'" = 0. 
Eliminating the coefficients we get 
or 
1 
< l 
g 2 
g 3 . . 
. . g*~ 2 
g*- 1 
g" 
1 
Sg 
Sg 2 
Sg 3 . . 
. . S q k ~ 2 
Sg"" 1 
Sg" 
0 
1 - AT 1 
Sg 
Sg 2 . . 
. . Sg"- 3 
Sg" -2 
p 
0 
1 - 2k- 1 
Sg .. 
. . Sg*“4 
Sg"" 3 
Sg*- 2 
0 
0 
0 
0 . . 
. . 0 
hr 1 
Sg 
l_ 
3! W 
S q S g 2 Sg 3 
2k- 1 S q Sg 2 
0 Ar 1 Sg 
Sg Sg 2 Sg 3 Sg 4 
1 /gy-4 3A;- 1 Sg Sg 2 Sg 3 
+ 4!W 0 2 At 1 Sg Sg 2 
0 0 k- 1 Sg 
V. 
r 
/ 
(47) 
(48) 
With a multiplex of the 4th order we find from the above that Av= — v 
in the cases when 
R — Q> Ql-iHb’ eq. 
Hence in (12) above we may put R instead of K with any one of these 
meanings. The condition that (g — a?) -1 may be infinite is that the denomi- 
nator on the right of (12) is zero, when in that denominator we put R for 
K and q — x for q. This namely, 
(g - jc)R(g - x)[(q - aj)E(g - x) - 2S.(g - ar)R(g - x)] = 0 . (49) 
which is a quartic in x, must by linity theory be the quartic of q. The 
coefficient of each power of x must, of course, be a scalar. If we substitute 
q for x we get another form in the present case for (47) or (48). I have 
not been able to generalise this to a multiplex of any order. 
In conclusion, I will return to the matters discussed in the first complete 
page of § 11 above. I no longer (April 1908) have doubts as to the sign of 
l 2 to be taken as a standard, but my present views are not correctly 
described on that page. 
I think we ought definitely to accept the convention that for a real 
fictit i 2 = —1. [We may still call i an imaginary fictit when i 2 — +1.] 
