583 
1907-8.] Algebra after .Hamilton, or Multenions, § 11. 
the (n — l) th order multits, when n is odd, are all of even order; and the 
multenions formed from them have precisely the same properties as those 
of what I have called the continent multiplex ; so that there is absolutely 
no reason why we should ever impose artificially the condition that the 
product of n independent fictits is a scalar. The product of the ( n — l) th 
order multits just mentioned is of necessity a scalar. 
I did not realise these simplicities when writing the paper ; but with 
the light that has come from later reflections, I have no hesitation in 
recommending these two important restrictions, namely : a real fictit shall 
be one for which i 2 = — 1 ; and (2) there shall be no continent multiplex. 
I am inclined also to restrict the meaning of replacement to what is 
above called a unireplacement. What is called above the fictorlinity 
replacement may very fairly be called a strain; e w ( )e~ w a fictor rotation; 
q{ )q~ l a multenion rotation ; and what is above called a complementary 
replacement may be called a complementary substitution. 
Geometrical applications are not our object here, but I may just state 
that I find there are interesting Euclidian-geometry applications of 
quadriquaternions. 
A quaternion may be looked upon both geometrically and algebraically 
as a degenerate octonion, an octonion as a degenerate Combebiac tri- 
quaternion, a triquaternion as a degenerate quadri quaternion. A quadri- 
quaternion is defined as 
io p + q + /xr + coSj 
where p, q, r, s are four independent quaternions ; co,' p., w are commutative 
with quaternions and satisfy the equations 
/x 2 = 1 , a) 2 = a/ 2 = 0, 
O)(o' = 1(1 - fx), C 0(0 = J(1 + /x), 
0 OfJL — (O — — fXCO , — (O fX = 0) = [XU). 
A triquaternion requires p — 0, an octonion requires p = r = 0, and a 
quaternion requires p = r — s = 0. 
A quadriquaternion may be regarded as a multenion of a fourth order 
multiplex in many simple ways. The following is one. Let 
, 2 — . 2 _ . 2 — . 2 — 1 
— l 2 L Z ~ l i ~ 1 J 
and let the i, j, k of quaternions be given by 
^ = l 2 t 3’ j = l 3 l V k = L i l 2 • 
/X = 
(0 = 1^(1 — /x)t 1 t 2 t 3 = 2 t l t 2 t 3(^- "t V-)> 
u = + /d W3 = 
Then we may put 
