527 
1907-8.] Algebra after Hamilton, or Multenions, §§ 5, 6. 
whether q c be wholly, in part, or not at all within the multiplex a 1 a 2 . . . a n - 
This I noticed when treating of the fictor linity replacement (§ 8). 
Putting n = 2, (14) becomes the second of (7), § 4; so we may say that 
(15) is the “satisfactory generalisation” then sought, but not found. 
6. A Multenion in terms of given Fictors. — In this section certain 
restrictions as to notation will for clearness be strictly adhered to, and the 
reader is advised to study carefully the preliminary explanation now to be 
given. 
n fictors of a group will be denoted by a_ x , a_ 2 .... a _ n ; by /3_ 1} 
P - 2 fi-n ; by «_!, /3_x, etc. 
Positive suffixes will indicate the order (e.g. aj, c) = S c a (c) )- 
An index dash „ „ „ number of fictors in a fictor product 
(e.g. a {c} is a product of c a’s). 
The sequence of fictors in a fictor product is, for the purposes of this 
section, frequently of vital importance. We bind ourselves in this matter 
by no rules of notation. 
The following theorem [(1)] is true whether a_ 1} a_ 2 , .... a_ n , n given 
fictors belonging to a given fictorplex of the n th order are or are not 
independent. It is also true of the continent fictorplex if the summations 
refer, as usual, only to the odd orders, or only to the even orders, q is any 
multenion of the given multiplex. Let a (c) be any product of any number 
c of different fictors a_ l7 a_ 2 , .... and let a ( c c) = S c a (c) . Let a <n ~ c) be the 
product of the rest of the fictors arranged in such a sequence that the 
(combinatorial part) S n a (c, a (w_c, [ = S B a?a't c c) ] h as the same value for all 
values of a (c) . Then 
2S n a< c W”- c > = %^$ n qa%I? > 
=2#s fl %] • • • ■ {) 
Here 2 2 rather than 2 is used to imply two summations, a summation 
within a given order c, and a summation of the different orders. In the 
present section S will be used for either of these summations singly, and 
2 2 will imply that both have to be made. 
The most important case of (1) is when a_ x .... a_ n are independent. 
In this case S n a (c) a (n_c) is not zero, and (1) is for most purposes more 
conveniently written 
q = S^S^a^.S-^a^. ) 
= ^ 2 a?S.^[a^- c >S- 1 a (c) a (n - c) ] ) ' ^ 
Here, no convention about the sequence of the fictors in the fictor products 
is necessary. In the last form of (2) S" 1 ( ) has by (7) § 5 been passed 
into S n ( ), and therefore the last S n has been changed to S 0 or S. (1) may 
