515 
1907-8.] Algebra after Hamilton, or Multenions. § 4. 
is of order b, that is, a a v b a a ~ x is of order b. Since by (10) § 3 a a ~ 1 = a a X a 
non-evanescent scalar, we have to show that a„a 6 a a is of order b. Let 
a a = XL + x'l + .... + yX + y'X 
where i, X, X' .... are fictits ; and where i, i ... . occur in the 
product v b , whereas X, X' .... do not so occur. Passing the second a a of 
a a v b a a across the b fictits of v b we get 
O-aVb^a = ( - + ....+ yX + . . . .)(xl + . . . • - yX - . . . .)v b 
= ( - y~\i 2 ^x 2 - i 2 2y 2 - 2 txyLX)v b . 
Here i\v b is of order b, since i occurs in v b and X does not. Hence 
a a v b a a is of order b. 
When q is a fictor product its reciprocal may be regarded as known 
by (10) § 3, and 
K a 2 .... <vT 1 = a a - 1 a a _f 1 .... oq -1 . 
This reasoning applies to the continent multiplex as well as to any 
other. This suggests a slight generalisation of the statement that q( )q~ x 
is commutative with S a when q is a fictor product. The proof I leave to 
the reader, as the statement is not subsequently required. If n is even, 
tu behaves like a fictit ; and if n is odd, like a scalar. Instead of 
supposing oq, ct 2 , .... to be mere fictors, then, we may suppose each a 
to be a fictor + any multiple of when n is even ; and to be of the 
form S 1 ^ + S w _ 1 g when n is odd. Then q( )q~ l is commutative with S a 
when q = oqcq . ... a b . 
It should perhaps be remarked that in our present subject a multenion 
is not in general a fictor product, whereas a quaternion is always some 
vector product. Many of the properties of quaternions therefore suggest 
properties that are not true of multenions in general, but are true of fictor 
products. 
In the above expressions involving a a , change a a to a. It is obvious 
that 
(xl + xl + . . . .A^S^cu^, ( yX + y'X+ . . . .)v b = S &+1 au 6 . 
Hence if q b be a multenion of order b, 
( - ) b aq b a = a 2 q b - 2aS b _ l aq b = - a 2 q b + 2aS &+1 ag & ^ 
= o?q b - 2S & _ 1 g & a.a = - a 2 q h + 2S &+1 ^a.a j 
Putting q b = /3 where /3 is a fictor so that 5 = 1, 
— aj3a = a 2 ft — 2aSa ft = — a 2 ft + 2aS 2 a ft. 
Thus (6) is a satisfactory generalisation of two well-known quaternion 
