1888 - 89 .] Rev. M. M. U. Wilkinson on Scalar Relations. 785 
relations, contained in the following 10 equations, five of which are 
easily obtained from the other five. 
e i e 3^5 = 
^1^3 e 5 > 
e i e 2, C o ~ C l C 2 e 5 > 
e 2 e 4^5 = 
^2^4 e 5 i 
C 2 e 2pA ~ e 2^3 C 4 } 
C 2 C 3^5 = 
^3 C 5 ) 
C 1^3 e 4 = e i C 3^4 3 
e 3 e 4 C 5 = 
C 3 C 4 e 5 > 
^l C 2 e 4 = e i^2 C 4 ) 
^A C 5 = 
C 1 C 4^5 > 
^i e 2 C 3 = C 1^2 e 3 * 
20. It can be readily shown that the expression for W 2 cannot 
be square rooted so as to express W in the form 
S( - 1 ) r Sa/3S/3ySySS8eSt£S£a . 
For a single permutation of any two letters changes the sign of 
W, while the successive permutations of a, ft; a, y; a, 8; a, e; a, £, 
being 5 in number, do not change the sign of 
Sa/3S/?ySySSSeSe£S£a. 
So the expression for W would not change its sign for a permuta- 
tion which would change the sign of W. 
E. Formulce for Sa/3y, &c. 
These may be obtained in various ways. Thus we have 
SaySSae£ = a 2 SySVe£ + SaySaSV£e 4- SaSSayVe£ • 
SaycSa^S = a 2 SycV{8 + SaySaeVS£ + SacSayY^ ; 
Say£Sa8e = a 2 Sy£VS € + SaySa^VcS + Sa£SayVSe ; 
whence 
SySSe^SaySSae^ + Sy € S£SSay<rSa£S + Sy£SScSay£SaS€ 
= SySS€£(Sa£Sa<$ye - SaeSaSy£) 
+ Sy€SS£(SaSSa€y£ — Sa£Sacy8) 
+ Sy£S8c(Sa€Sa£yS — SaSSa£yc) = — F 2 . . . (41) 
as appears from (26). 
We have also 
SaySSac^ + SaycSa£S + Say£SaSc = 0 , 
