537 
1907-8.] Algebra after Hamilton, or Multenions, § 7. 
(23) gives h {c) , and therefore the n- tic in terms of any n independent 
hctors. This is one method familiar to qnaternionists. Another is to 
express h {c) in terms of the a, /3 of (pp = 'Z/3Sa\p [(4)]. This is at once given 
in terms of (12) and (20) thus, 
= % S$ C VP (25) 
h {n) is the familiar discriminant. This is of such fundamental importance 
that I propose to reserve the * symbol [0] for it, the vertical lines of the 
square brackets being intended to recall the vertical lines of the deter- 
minant form. Thus 
[0] = ¥ n l= \(<f>v) ( n\ = S n 0a 1 0a 2 . . . . cf>a n S~\a 2 . . . . a n ) 
= S«0q0t 2 .... 0t n .(t 3 t 2 . . . .t n ) _1 i 
With this notation the scalar n- tic [(21)] itself is 
0 = [0 - x] = W c) ( - x) n ~ c (27) 
and the 0 w-tic is obtained by replacing the scalar x by the symbol 0. 
To get the reciprocal 0 _1 of 0 put in the eq. [(26)], 
[0]S n a 1 a 2 . ... a n — S n 0cq0a 2 .... 0 a n 
a 1 = p, a 2 = rj v . . . .a n = Y\ n _ x ; multiply by and apply (29) § 6. Thus 
p[0] = K^fr 1 1) S w (0?7) (n_1) 0p (28) 
or when [0] is not zero, 
0-v = ) ( 
= [0]-^(00)»s n 0- i 0{ • • • ■ v ; 
the last form being given by (13). These express 0 _1 explicitly both in 
terms of 0 and of 0', at any rate if we add 
M = M (30) 
which is a particular case of the statement that on changing 0 into 0' 
h {c) is unaltered [(20), (13)], that is, the two r^-tics are identical. It also 
follows from these results that 0 _1 and 0 ,_1 are fictorlinities and are con- 
jugate to one another. 
We also have 
[00] = [0][0] (31) 
which contains the usual theorem for the product of two given deter- 
* There is here a most unfortunate oversight in the notation. In § 8 below, a series of 
multilinities 0 O , <p v <p 2 . . . . <p n of a quite fundamental kind are defined in terms of 
<p ; (p l = (p and 0 C is of c dimensions in <p. The oversight is that in this series <p n is the scalar 
here denoted by [0]. 4> n is a far more expressive symbol than [0], and should, throughout 
the paper, be read instead of [ 0 ]. — [ Note added April 1908.] 
