132 Proceedings of the Royal Society of Edinburgh. [Sess. 
(12) is a quadratic expression in a. It cannot vanish for more than five 
arbitrary values of a unless it vanishes identically. For any scalar 
component of (12) may be regarded as defining by its vanishing a quadric 
cone through six arbitrary vectors f3, y, S, e, p, cr, with variable a. But 
this is impossible, that is, each component of (12) vanishes identically. 
6. Methods for deriving New Identities. 
An almost unlimited number of new identities may be obtained from 
(12). Among the more obvious methods we may consider the following 
three : 
1. Giving particular values to the seven vectors, and special forms 
to the function F. 
2. Acting on (12) by various differential operators. 
3. Replacing one or more vectors by the operator V- 
These methods I shall now briefly exemplify. 
7. Identities obtained from (12) without Differentiation. 
As a first example let F(a) = a 2 . Then 
a 2 Ca + ^ 2 C/3 + 7 2 C 7 + . . . o- 2 Ccr = 0 . . . (13) 
or briefly 2a 2 Ca = 0. This is, of course, a scalar identity. 
Again, let F(a) = aScrr, where r is any vector whatever. This gives the 
vector identity, linear in r, but quadratic in the other seven vectors, 
aSa/rCa + /3S/3rC/3 + . . . + (rSa-rCcr == 0 . , (14) 
or 2aSarCa = 0 where the summation is with respect to the seven vectors 
of (8). 
Again, the symbol S in (14) may be replaced by Y, giving 
CaV a Y ar 4- C/3Y /3Y j3r + . . . + CcrVcr V'ar = 0 . . . (15) 
To the vector t occurring in (14) and (15) special values may be given* 
Thus if r = Y pc we have from (14) 
aSapcrCa + /3S/3pcrCf3 + ySypcrCy + SSSpcrCS + eSepcrCe = 0 . . (16) 
for the last two terms of (12) now vanish. 
Multiplying (16) into Se and taking scalars gives 
SaSeSaptrCa + Sj8SeS/3p(rCj8 + SySeSyprrCy = 0 . . . (17) 
a scalar relation connecting the three expressions Ca, C/3, and Cy. In a 
similar manner we may obtain these two other relations which involve 
the same C’s : 
SaSpSaecrCa + S/3SpS/?ecrC/3 + SySpSyecrCy = 0 . . . (18) 
SaSo-SaepCa+ S/3SaS/3epC/3 + SyScrSyepCy = 0 . . (19) 
