536 
Proceedings of the Royal Society of Edinburgh. [Sess. 
times only, b occurring B times only, etc., the given fictorplex consists of 
the following independent fictorplexes, 
(A.J X 2 , .... X A ), (/X 1} /X 2 j .... ^b)j • • • • 
[or say (A) , (/*), . . . . ] 
which are such that 
0A X — $ A l5 0A 2 = cl\<i + X l5 . . . . , 0A a = <xA a + X i 
<f>fh = bpi, = bfa + /x' 1? . . . . , 0/x B = bfi B + /x' B _i > • • (22) 
where X e stands for some fictor of the fictorplex (X 1} X 2 , . . . . X e ), and 
similarly for fj! e , etc. Among other things (for which see the chapter) it is 
further shown (1) that the independent fictorplexes (X), (/x), .... are 
definitely determined by 0 ; (2) that (0 — a) A kills every fictor of the fictor- 
plex (X) and no other, (0 — bf kills every fictor of (jm) and no other, etc.; 
(3) and therefore that 
(0 - a) A (0 - b) B . . . . = 0 n - /i'0"- 1 +....+(- ) n h {n) 
kills every fictor of the given fictorplex of order n. [Note that if 0 be 
supposed of such form as to be capable of operating on fictors not belonging 
to the given fictorplex, then even if we impose the condition that 0 kills 
every fictor outside the given fictorplex, it is not true in general that 
0 n — . . . . + ( — ) n h (n) kills them. In this case we have to raise the degree of 
the quantic by 1, and say that 0(0 n — /i'0 n_1 + . . . .) kills every fictor 
whatsoever.] 
To deduce the form of h {c) in (20) from (21) we have 
^ = ^S n (0a)( c >a (w - c )S-V c) a (n - c ) .... (23) 
ft (c) = 2 S.(0a)< c fa^- c )S- 1 a ( V n - c >], [(7) § 5] 
= ^S.(0a)S|? [(7) §6] 
= S.(0^V [(19) §7] 
We get a second standard form of h {c) from (23) thus, 
U c) = S n (0a) (c W n_c ]S n d^d (n_c ) [(16) § 6] 
whence by (19) § 7 
0 n -/i'0 n ~ 1 + + (-fU n) = 0 
where h (c) = (c !) _1 ([w - c] !) _1 S n (0^) (c) ^ (n “ c) |S„^ (c) ^ (n_c) 
= tt c c .s(0#v^iM c V n -^ 
These two forms of h (c) are given to prevent ambiguity as to the con- 
nection between y and f here understood. [For the full quaternion forms 
of (20) and (24) see Utility of Quaternions in Physics, text and footnote, 
p. 17.] 
