1888 - 89 .] Rev. M. M. U. Wilkinson on Scalar Relations. 775 
we have the important formula, 
Sa/?yS8e£ = 
Sa£ , Sac , SaS 
S Pt, S/?c, S/38 
SyC, Syc, Sy8 
of which (1) is a particular case. 
Also, since, 
. . (5) 
Sa/3y8c£ = Sa/?ySSc£ + Saygy(8Se£ - cS8£ + £S8c) ; . (6) 
we see that the expression for Sa/?y8c£ in terms of Primary Scalars 
contains 6 + 3x3=15 terms in all. 
6. Representing the 10 determinants (of which Sa/?yS8e£ is one) 
as follows : — 
Sa/}yS&£ = V ; 1 
Sa/3fSySc = A.?V; Sa/?6Sy£S = A!V; Saj3SSy4 = \%V ; I 
SyafSySSc^/xfV; SyaeS y8£S = /4V; SyaSS^ef = tfY ; ’ ^ 
S/3y£SaSe = yfV ; S/3yeSa£S = v^V ; S/3ySSae£ = VgY ; _ 
we have, at once, 
1 = A2 + XI 4. A.2 = y^2 + y^ + ^2 __ v 2 + „2 + v 2 __ | 
= A-f + p-1 + Vi = A| + /i| + V 2 = + fA + v \ ) i 
Now define as follows, 
W = SaycSaS£S/3e£S/3y8 — Say8Sae£S/3S£S/?yc ; . . (9) 
a little consideration will show that, by permuting the Vectors, there 
are only two expressions of the form W, and that W 2 is a sym- 
metrical function of the Vectors. Thus, since, 
Sy3y8S/?c£ + Sy8yeSy8£8 + S/?y£S/?8c = 0 ; 
Say8Sae£ + SaycSa£8 + Say£Sa8c = 0; 
we have, 
Sy8y8Sy8e£SaycSa£S — Say8Sae£S/?ycS/3£8 
= Say£Sa8cS/3ycS/?£8 - Sygy£S/38eSaycSa£8 . 
Calling the permutation of any two Vectors one permutation, an 
odd number of permutations changes the sign of W merely. The 
formulae (7), (8), readily show that W 2 is symmetrical. 
