543 
1907-8.] Algebra after Hamilton, or Multenions, § 7. 
given (58), then y' = — y, and that given x = — X> then y involves \n{n— 1) 
arbitrary scalars. 
From the definition Sojyp = SpjyV, %'= — X we g e t by putting o- = p 
that 
Sp\ x p = 0 (59) 
for every p i This also follows from (58). 
Conversely if (59) holds for every p, y is skew. For = 8>* 2 |yf 2 = 
.... = 0 ; and from 
Sfah + yt 2 ) \x( XL i + Vhi) = 0 
it follows that Sijy^ — — S^lytj ; and from this it follows that 
^/°lx cr= — Strlxp for every p and cr; or y' = — y. 
The fictits i v i 2 . . . . of (56) may be called the undeviated fictits of the 
colinity <p. y'y = — y 2 is a positive colinity, so that y 2 is a negative 
colinity. Let y\ = — a 2 i, and first let a be zero. Since yp = 0 if, and only 
if, y yp = 0, x i i is in the present case zero. 
Next let a be not zero, and put ar 1 y = y 1 , and Xi l i = a - Then 
Xi 2 h= ~ L v Xih = a > Xi a= Xi 2a= “ a - 
From SiJxj^ = 0 and S^ 1 |y 1 a = — Salyp-L we get 
St 1 |a=0, Sa|a = St 1 |t 1 . 
Hence we may take a — i 2 ‘, and we see that the root — a 2 (when not 
zero) must occur at least twice in the y 2 w-tic. If it occurs more than twice 
we have now only to take i s in the fictorplex of undeviated fictors of y 2 
corresponding to this repeated root —a 2 ; and the same reasoning will apply 
to as to i r 
Therefore when y is skew, there always exist n fictits i v i 2 , ..... i n such 
that 
= <%, yt 2 = - ai v yt 3 = &t 4 , yt 4 = - &t 3 (60) 
and when n is odd, y* n = 0. Thus with the notation of (55) we have 
0 
a 
0 
0 . . 
- a 
0 
0 
0 . . 
0 
0 
0 
b . . 
0 
0 
-b 
0 . . 
In the notation of our present subject the following is the appropriate 
form, 
XP = Sjwp where w = ai 2 ij -1 + fri 4 i 3 -1 + . (62) 
According as n is even or odd the ^-tic is 
(x 2 + « 2 )(x 2 + ^ 2 )... • I 0, or x(x 2 + « 2 )(x 2 + ^2 ) . • • . =0. ■ (63) 
