774 Proceedings of Royal Society of Edinburgh. [sess. 
All Scalars, it is clear, express in terms of Tensors and Primary 
Scalars. 
Thus, when n — 4, we have one such independent equation, as 
a 2 , 
Say8, 
Say , 
SaS 
= 0 . . . (2) 
Safi, 
p , 
S/ 3 y, 
S£S 
Say , 
S/8y, 
r 2 - 
SyS 
SaS , 
S/3S, 
SyS , 
S 2 
Equations such as 
SaSSa/?y = a 2 S/?yS + Sa/3SyaS + SaySa/3S ... (3) 
are not independent equations, as they can be obtained from (2) by 
means of equations of the form (1). 
4. In the case of n = 5 there are three relations connecting the five 
Tensors and tenTPrimary Scalars. Here various problems present 
themselves, of this character ; having given twelve of these Scalars, 
to find equations connecting them with the other three. Of course 
the twelve must be so selected as not to contain ten which are 
functions of only four vectors, and which would, therefore, be con- 
nected by an equation (2). 
In the case of w = 6, we have six Tensors and fifteen Primary 
Scalars, connected by six independent equations. So if fifteen of 
these, so selected as to be a set of fifteen independent Scalars, are 
given, six equations sufficient to determine the remaining six, can 
be found. 
The problem we aim at discussing is, in the case of six Vectors, 
having given the fifteen Primary Scalars to express in terms of 
them the Tensors, and other Scalars. 
B. Principal Formulae. 
5. Our Vectors are a, j3, y, S, e, £. 
We have at once, 
Sa/?yS = Sa/?SyS - SayS/38 + SaSS/3y ; . . . (4) 
and, since, 
Sa/?yS Se£ = SajG(VScSy£ + Ve£SSy + V£SSye) ; 
