346 Proceedings of Royal Society of Edinburgh. [sess. 
Take any two vectors i, j, and consider the expressions 
S .ufyj = SJGX; + S.iej 
S .j<j>i = S.jtti + S .jei 
S i&j is the same as S j&i because T3 is self conjugate. But S iej and 
Sjei are of different signs necessarily. Hence if S .i<f>j and S .jcf>i 
are of the same sign, neither S .iej nor S.jei can he greater than 
Si&j. 
The remaining expressions treat of bi-vectors, and their inten- 
tion is to extend the investigation to the case where <f> has only 
one real root. The root-vectors and the corresponding root values 
are 
a , p. + ic, p - 1€ 
aI1(i 7-7 7 -7 
g , /i + ik, h - ik 
i being now the symbol for J — 1 and ghk scalars. The expression 
for D.<£p becomes 
<j6/oSa(/? + ie)(/3 - ie)=gaS(/3 + ie)(/3 - ie)p 
+ (h + ik)(P + ie)S(f3-ie)ap 
+ (A - ik)(/3 - ?’e)Sa(/3 + ie)p 
which reduces to 
c t>pSae/3 = ga$e/3p + (he + k/3)B/3ap + (h/3 - ke)Saep . 
This form has already been partly discussed in the paper of 
June 7, 1897, and lines of development are indicated in the earlier 
paper of March 1, 1897, with which indeed these notes seem to 
have a very close connection. 
( Issued separately January 17 , 1903 .) 
