558 
Proceedings of the Royal Society of Edinburgh. [Sess. 
In (7) let n be odd ; divide the summation into two parts for which 
c = 2a and e = n — 2b where a and b are positive integers ; then without 
restricting q to being of even order we have 
] 1 
'(26)1 j 
(35) 
(30) 
q = S‘dS 2a ^- 2 «lS,X^.S 2a /3< 2 «)).S-^< 2 ^^- 2 «>] 
+ 2 2 [S 2& ^ (26, .S(g^^H| 26) ).S _1 ^ (n_2& ^ 6) ] 
Here no convention as to the sequence of the /3 factors is required. 
When q is of even order S n (g.S 2a /3 (2a) ) is zero, and we get 
= 2 2 S 2& /P 6) .S (q. S 26 /P*- 2&) ) 
Here, of course, the convention is required that the sequence is such that 
S/ 3 (n_ 2 ® (2& ) has the same value for all values of b. 
(36), which is true whether f3 v /3 2 , .... are or are not independent, 
develops q in terms of the products /3 {c) of an even number of factors. It 
may be regarded as the natural simplest generalisation of 
pSafiy = V (By Sap + V yaS f3p + V afiSyp. 
The corresponding generalisation of 
pSa/3y = aS /3yp + (3Syap + ySa /3p 
may be obtained by changing q of (35) to qxss, and supposing the new q to 
be of even order. Thus by (7) § 5 
q$/3 (2a) /3 {n ~ 2a) = % 2 S 2a /3 in ~ 2a \S(q.S 2a fi {2a) ) . . . (37) 
It seems scarcely necessary to remark that C S C /3 (C) is a combinatorial part 
of /3 (c) with reference to the factors /3, because S c a (c) is one with reference to 
the factors a. 
All that it is necessary to remember about (36) and (37) is that q can 
be expanded in the form 2«S 26 /3 (2&) and also in the form 2h/S 2ai 8 (w_2a) . The 
coefficients are easy enough to obtain even mentally. 
If we put 
w 0 = armors), . 
(38) 
we obtain several convenient ways of treating p the general multenion of 
an odd order multiplex in terms of two even order multenions, q, r of the 
multiplex. Thus we may put 
p = q- f£Xr, or instead q + T, T 0 r, or q+JZp', or £q + r/r . (39) 
where 
i=J(l+W 0 ), , = |(1-CT 0 ) \ _ _ _ uo) 
and therefore £ 2 — £, rf — y, — 0 — rj£, £ + rj = 1 j 
Here tu, and therefore vj 0 , £, tj, are all commutative with p. Thus 
,q + js 1 r is the analogue of Hamilton’s bi-quaternion. 
