510 
Proceedings of tlie Poyal Society of Edinburgh. [Sess. 
and in particular 
S.gK# = 3x 2 (18) 
Also 
K(qKq) = qKq, QfeQg) = qQq . . . . (19) 
and similarly for uni-retroplacements in general, but not for uni- 
proplacements. 
P, Q, and K and their powers and products are all commutative with S a , 
that is, 
K(S a ?) = S 0 (Ky), etc (20) 
Also 
= )°S tt g (21) 
Thus J(l + P)g contains the even order parts and J(l — P)g the odd order 
parts of q. Similar statements cannot be made about Q and K till certain 
permissible simplifications are added to our fundamental laws. 
The following list of notations habitually used below may prove of 
service : — 
a, b, c, x, y , z will usually denote scalars. 
l 15 l 2’ ' * ■ ' ’ L j • • • * ? 1 
V, V , v a , v b „ 
ft 7> a n 
P, <b r , 
^b 5) 
„ fictits. 
„ multits (v a , a th order). 
„ fictors. 
,, multenions. 
,, fictor products. 
„ a fictit product. 
3. Permissible simplifications. — We shall but rarely refer to the 
continent multiplex, which has been introduced mainly to call attention to 
the peculiar explicit symmetry of quaternions. 
Unless the contrary is stated, we shall always suppose our symbols to be 
confined in meaning to a given multiplex of order n. A quaternion is 
still a multenion, namely, one for which n = 2. The Yq of the quaternions 
is our present S l q-\-S 2 q, and consistently with our fundamental laws we 
have for quaternions to assume 
fi 2 = ‘2 2 = -!• 
The i, j, h of quaternions may be identified thus with our present 
symbols 
i — q, j — i 2 , h = ip 2 , 
from which the reader may verify that 
i 2 —j 2 = k 2 = ijk = - 1 
i=jK j = hi, h = ij. 
