519 
19 07-8. J Algebra after Hamilton, or Mnltenions, § 4. 
Changing the sign of A c we get for the new value of q c 
- Kq c -\ L c~ l + 9.0-n 
and this can only be zero if q c _ x — 0. 
Suppose now we have determined q' so that 
9 fc i = — A^ 9 1 •••• i 9 L a = ~ A a9 1 9 L ct+ 1 = '"t A a+l9 i • • • • » ^ = + A n q . 
If a is even put q = q'i p 2 . ... i a ; if a is odd and n — a is odd put 
q = q'i a +i if a is odd and n — ct is even put q = q'i a+1 .... tn _ 1 . We 
then have in the first two cases 
9^1 = A \9j 9^2 = * 5 9^n — 1 — ^n—l9i 9 L n = A n 9i 
and in the third case we have the same, except that the last relation is 
replaced by qi n — — \ n q. This want of harmony in sign, it will be noticed, 
only occurs when n is odd. And even this can be avoided if we choose to 
interchange A n _j and \ n instead of changing the sign of \ n ; or instead we 
may change the sign of every A [operate by A n ( )A n -1 ] instead of a single 
selected one. This change of sign (of one or of every A) will be called 
perversion. 
These results obviously follow from the relations 
h*h*h — h* ^ == > etc. .... ( 18 ^ 1 / 
And that we can interchange two at the same time as we change the sign 
of one only, follows from 
(1 + 1 1 L 2 ~ 1 )l 1 (1 + = - 1 2> (! + t l t 2 _1 ) t 2( 1 + L l L 2~ l )~ 1 = L V 
(l + ^2" 1 ) t 3( 1 + t l t r 1 )~ 1 = t 3* 
The question remains whether q~ l is finite; and although I cannot 
answer this, I think the following remarks may help others desirous to 
clear up this point, q must satisfy (12), and therefore it must satisfy (15), 
as may be proved from the general theory of multilinities. We have then 
only to consider the question whether with q given by (17) [where (a) r is 
arbitrary, (b) 0, (c)\ls 2 = \Js (so that \J/ q = q )] for some value of r, q~ x is 
finite. It seems to me that some very special condition would have to be 
satisfied by \ v A 2 . . . . A n to render all such values of q _1 infinite. At the 
same time such a condition is 
AjA 2 .... A n = scalar, 
which requires that n is odd. That this condition does render g -1 = oo for 
all values of r appears thus : We have seen in § 2 that the condition may 
be satisfied by 
