580 
Proceedings of the Royal Society of Edinburgh. [Sess. 
We will now find in terms of c, the number of symbols I a , I & , .... in 
the equation 
[c = number of operators I 0 , I & , ....], K// = v Aqv* 1 = Q T a I & . ... q . (45) 
the condition that Av — — v. The interest of this lies in the fact that when 
Av = — v and p = Ry> (for instance when p = gRg) then p satisfies a simpler 
equation than usual, that is an equation of degree 2 m_1 instead of degree 
2 m as usual. 
Let I a I & .... contain x of the symbols l v I 3 , 1 5 , . . . . and y of the 
symbols I 2 , I 4 , I 6 , . . . . so that c = x+y. Putting each I as t3l{ )r 1 tcr“ 1 we 
have 
v = product of ( x of 6Tq, Wt 3 , . . .) and (// of di 2 , £ 7t 4 , ....). 
Hence noting that 
A(t«7t 4 ) = — T77t 4 , AipSt^j — + 67i. 2 
we obtain 
Hence Av = —v when c is m+ 1, or m-f-2 or any integer differing from 
either of these by a multiple of 4. In other cases, of course, Av = -\-v. 
When R = Q, c = 0, and when R=PQ, c — 2m. Hence 
For R = Q , Av = - v , we must have n = 2m = 4, 6, 12, 14, 20, 22, . . . . ) 
„ R = PQ, Av= -v, „ „ n = 2m — 2, 4, 10, 12, 18, 20, .... J (4 } * 
Multiples of eight occur in neither list, so that to obtain the lowering of 
the degree, in those cases, some other retroplacement than Q or PQ must be 
used. Other multiples of four occur in both lists ; (12) above is a case of 
this. 
If q is unrestricted let k = 2 m ; if q ■= Rg and Av = — v let k = 2 m_1 . Thus 
in both cases k is the degree of the identity satisfied by q. Also in both 
cases 
k$q c = sum of c th powers of the roots. 
This has already been stated in (C) above for the first case, and the second 
case is easily deduced from the first, thus. The Grassmann identity, when 
g = Rq and Av= —v, has roots which consist of pairs of equals, say a, a , 
6, b, ... . and the first case gives 
2 r,l Sq c = 2(a c +b c + . . . .) 
The roots of the lower degree identity satisfied by q in this case are 
a, b, .... so that 
yfcSg c = 2 m - 1 Sg c = a c + 5 c 4- 
= sum of c th powers of roots. 
