556 
Proceedings of the Royal Society of Edinburgh. [Sess. 
We could clearly here write any symbol X for tjr 2 , provided X 2 = l. Thus 
we might write any one of l 2 , P, Q, K. We thus find that 
( 18 ) 
Thus K m_1 is commutative with and if we pass either K or K m 
across m we must multiply by ( — ) n_1 tcT 2 . Hence 
B q = (P 0 + GTPjJK m q = K m -\F 0 + CTPJK q = K™[P 0 + ( - f-^PJg . (19) 
By aid of the definition of B' the conjugate of B, namely 
SgKBV = SrKBg, 
we now have 
Hence 
B'g = P 0 K m g + P^ErP” X K m q) 
• (20) 
\n even] 
= (I*o — ) k “2 = PEs .... 
• (21) 
[n odd] 
B'g = (l +5T)P 0 K"g = K’"(l +5J“ 1 )P 0 g . 
• (22) 
\n even] 
B'B 2 = (P 0 -®P 1 )(P 0 -^ 1 P 1 )g = 2 . . . 
• (23) 
\n odd] 
B'Bg = (l +W)(P 0 + cn- 1 P 1 )g=(] +WP »~^)q 
• (24) 
\n even] 
B 2 g = Pg . 
• (25) 
1 B 2 g = ( P 0 + rar^PJg = K^-^Bg 
• (26) 
\n odd] < 
' BB' = 2P 0 , BB'B = 2B, B' 2 = (1 + 67)P 0 , B'BB' = 2B' 
• (27) 
<B 3 -B 
• (28) 
Of these the most important are the last three for n odd and the, here 
re-collected, still simpler corresponding ones for n even ; 
[n even] B'B = 1 = BB', B' = PB, B 2 = P, B 4 = 1 . . . . (29) 
Thus for n even B is a rotational multilinity, and may (a rather unexpected 
result) be said to be a square root of P. 
In the equation 
Bq = ( l 2 )J(n+ i )(n+ 2 ) trTti (30) 
we may suppose 
67 = qqq . ... c n . . . . . (31) 
Supposing now an exactly similar replacement is made again, what is 
the new value of q ? that is, what is the value of 
[(Bq) 2 ]^ +1 >< M+2 >B67.Bq ? 
It will be found that it is — q when n is even, and it is Bq when n is 
odd. Thus when the replacement is made twice or several times exactly 
according to (30), (31) we get successively for q 
\n even] g, Bg, P q, PB q, q, Bq, . . . . 
\n odd ] q, Bg, Bg, Bg, .... 
Let us call the case when n is odd and tsr 2 = — 1 “ anomalous ” ; when 
