512 
Proceedings of the Royal Society of Edinburgh. [Sess. 
Since N is odd, it follows that if the scalar z?y N is to be real, 
Hence we put 
t i 2==t 2 2 = = (- 1 ) 2 .... (6) 
The sign of the scalar is still left arbitrary, and no inconvenience 
results in leaving it thus, at any rate for the present. [Quaternion 
assumption is in harmony with either 1 or = q 2 . But it must 
be remembered that the sign of -ny N depends on the sequence of the fictits 
in it, and we have said nothing in our fundamental laws about such an 
ordained sequence.] 
The following results for any two fictors a, a may be here noted. Let 
a = flqq 4" 4" . . . . 5 a = x -j q 4- x ^ q 4- . . (7) 
then 
aa = - l 2 (£q£C j 4“ XcflC 2 4“ • • • • ) 4" 2 — j) 4" .... . . (8) 
Hence aa contains only zero th and second parts (to be easily generalised 
later), and 
aa = — a a 
aa! — a! a 
If q be a multenion of order a we have 
Q &» = (“) 
a(a— 1 ) 
~2 . 
i, S 2 aa = — Sga’a 
• (»> 
: Sa 2 — scalar .... 
• (10) 
aa ! , aa! — a! a = 2S 2 aa/ 
• (11) 
when Saa' = 0 ) 
• (12) 
when S 2 aa' = 0 J 
; we have 
a(a— 1) 
r «=(‘ 2 )“Q?«=(-) 2 (<■*)*% • 
• - (13) 
3 acc. as t 2 = ± 1 
• (H> 
Calling ^(l-fQ)g ^he Q 0 part and |(1 — Q)g the Q x part of q; and 
similarly for PQ ; we have from (13), on putting S a q = q a , 
K 1 + Q)2 = Qo# = % + <h + <h + % + % + • • • • [ 
J(1 - Q)2 = Qi^/ = g 2 + 2'3 + 3'6 + 3'r + 2'io + - • • • J 
i( 1 + P Q)^-( P Q)o3' = ^0 + ^3 + ^4 + ^ + ^8+ | 
K 1 - P Q)^ = ( P Q)i 2 = ?1 4- q 2 4- q 5 4- q 6 4- q 9 + 
For convenience of reference we here add [(21) § 2] 
|(1 4-P)g = P 0 g r = g 0 4-g'2 + 9'4 + 
i(! - P )^ = P l^ = ^l + ^3 + ^5 + 
(15) 
(16) 
:::} 
(17) 
Also (14), (15), (16) point out the K 0 and K x parts. The K 0 and Q 0 parts 
may appropriately be called the self -conjugate and self-reversate parts 
respectively. Thus by (19) § 2, qKq and qQq are self -conjugate and self- 
