523 
1907-8.] Algebra after Hamilton, or Mnltenions, § 5. 
To prove the theorem let a, a be two consecutive fictors of xn a . In 
S„a 1 a 2 . ... aa .... a a we may, in accordance with (3), write S 2 (aa') 
instead of aa, and by (9) § 3 the sign is changed when a, a are inter- 
changed. We may now again suppose the S 2 removed, and the theorem 
follows. 
That these simple and easily proved results contain really complex 
algebraic truths is evident from the following statements. The particular 
case of (3), 
S„[S 0 (a 1 o 2 .... a n )][s n _ ft (a a+1 .... a n )] = . ... a n ) . . (5) 
where a v a 2 . . . . a n are supposed expressed in terms of n fictits, is an 
expression of the theorem which develops any determinant of order n 
in terms of determinant minors of orders a and n — a. The equation 
S a+JZcP* = S a+& (S a ^ a S & rcr 6 ) (6) 
where a v ... . a a , /3 V .... /3 b are supposed expressed in terms of n fictits 
and n>a-\-b is the equivalent of a series of statements, of the same 
general nature, having reference to a rectangular array of (a + b)n 
elements. 
A complement of a multit of a given multiplex of order n means a 
product of all the fictits not constituting the given multit. If w is the 
product of all the fictits and v is a given multit we shall, consistently with 
this explanation, call each of the quantities 
ZJv, vTo, ZJ~ x v, vZJ' 1 
a complement of v; or, more generally, if q is any multenion of the 
multiplex, zoq, qu r, TZ~ x q, qv5~ x will each be called a complement of q. 
It does not seem desirable to restrict more definitely than this the meaning 
of complement, because these four forms are about equally useful. 
Putting, as usual, ^ = + + • • • . where q c — S c q, we have 
V5q = Tl T^q + + ZZq 2 + . . . . 
The terms on the right are of orders n, n — 1, n— 2, ... . respectively. 
Hence the first of the following, the others being proved similarly, 
— ^S n _ c g, S c qT3 = S n _ c q.TZ, 
S c ^lT ^q = CT *S n _ c ^, S c qZJ i 
i.e. we may pass tsj into or out of S c ( ) provided that at the same time 
we change S c to S n _ c . 
Since in the given multiplex there is but one independent multit of 
the n ih order, these equations are equally true if r n be substituted for w 
where r n ( = xvs) is any order multenion. Thus 
^ n^n— c(Zj n ®n— (fL'^n 1 /y\ 
&c r n~ l q = r~ l Q n _ c q, S c qr n - 1 = S n _ c q.r n ~ 1 ) 
