1919-20.] An Identical Relation connecting Seven Vectors. 135 
we may designate the vectors which define the quadric cone, in their cyclic 
order, instead of designating the omitted vector. Thus we might have 
written 
SA.V r . C(a/3ySep) = — L(a/lySe[Ap]) .... (25) 
where the vectors X and p may be interchanged without altering the value 
of either side. 
If we wish to expand the expression in explicit form we may return to 
(7) and obtain by actual differentiation 
L[(r, Ap] = L(a/lySe[Ap]) = Sya/lSSe/3(Sy3pSa€A + SySASaep) 
— Sy8/3Sae/?(SyapSSeA + SyaASSep) . . (26) 
which may serve as definition of the function L of seven vectors. 
With these conventions (24) would be written 
SarL[a, acr] + S/3tL[/?, /3crj + . . . + SprL[p, per] + ICtrScrr = 0 . (27) 
We may now obtain identities connecting the original six vectors of (4) 
by allowing cr to coincide (after the differentiation) with one of these, and 
also giving special values to r. If, for example, we let cr coincide with a we 
shall have C<r= — Ca by the former lemma, while L [a, aa-]=2Ca= — 2C cr. 
Furthermore, L [/3, /3cr ] becomes — L [or, a/3], and similarly for the other L’s. 
Hence (27), when cr coincides with a, gives 
2C(rSar = S/3 tL[<t, a/1] + SyrLjV, ay] + . . . + SprL[(r, ap] . . (28) 
from which cr is absent. By giving to r the value Yep this gives 
2CcrSaep = S/?epL[<x, a/3] + SyepL[(r, ay] + S3epL[(T, aS] . . (29) 
a useful identity connecting three L’s with one another. We note that 
these three L’s involve only the six vectors of the cycle (4). By letting r 
have the value V pa we should have a relation between four of the L’s not 
involving the quantity Co-. 
These last results were obtained by acting on (12) with Sv. Slightly 
more general results are obtained by acting with SAv, where X is any 
vector whatever. After the differentiation special values may, of course, 
be given to X, to the function F, and to the vector acted on by v. The 
number of identities thus obtainable is practically unlimited. 
A result which I have found of frequent utility in the general theory 
of quadratic vectors is obtained as follows: — Multiply (14) by aV and take 
scalars, giving 
S/lrS/laV . C/3 + SyrSyaV . Cy + . . . + S<rrS(raY . Ccr = 0 . (30) 
Now allow v to operate on < 7 , and we have 
S/3rL[/l, Y /3a, <r] + SyrL[y, Vya, cr] + . . . + SprL[p, Y pa, cr] + CcrScraT = 0 (31) 
