568 
Proceedings of the Royal Society of Edinburgh. [Sess. 
of h is I do not know. [This value ought apparently to be deducible from 
the great peculiarities of (7) and (8).] 
Up to n = 4 we have for any multenion q, 
(n< 5) 
K q(qKq-2S qKq) 
1 qKq(qKq - 2SqKq) 
( 12 ) 
the denominator on the right being a scalar, because ( qKq — SgKg) 2 is a 
scalar. Thus in the present case (n< 5) the Xg spoken of above is the 
numerator on the right of (12). 
Even when n = 5, q~ l can be explicitly exhibited although there are 
then 32 independent multits as opposed to the quaternion 4. Thus putting 
qQq=p 
since Q p =p, p contains no S 2 or S 3 parts. Putting its S 4 and S 4 parts in 
the form p + TScr where p and cr are fictors, it is easy to see that the square 
of S x p + S 4 j9 is a scalar + 2S 5 ( S 1 pS 4 p). Hence 
(p-Sp - S 5 p) 2 = scalar + 2S 5 (S 1 pS 4 p) 
or p 2 - 2p(Sp + S 5 p) = ^ + 2S 5 (S 1 2>S 4 p - SpS 6 p) 
where x is a scalar. Thus 
y = scalar = [x + 285(8^84/? - SpS 5 p)] [x - 2S 5 (S 1 2?S 4 p - SpS 5 p)] 
=p(p - 2Sp - 2S 5 p)[(i? - 2S p)(p - 2S 6 p) - 4S 5 .S 1 pS 4 p], 
or y = qQqiqQq - 2SgQg - 2S 5 gQg)[(gQg - 2SgQg)(gQg - 2S 5 gQg) j 
-4S 5 .(S 1 gQg)(S 4 gQg)]| . (13) 
[*< 6 ] q'^y-'Qqi „ „ )[ » „ ]‘ 
In (12) we may write PK, Q, or PQ in place of K. 
Similar to (12) and (13) we have 
[w<3] gPQg = scalar, g _1 = PQg/gPQg . . . . (14) 
This is, of course, virtually the quaternion case modified so as to be true 
when t 2 = +1 as well as when i 2 = — 1. 
