1919-20.] An Identical Kelation connecting Seven Vectors. 133 
The last three results, of course, involve the identical vanishing of the 
determinant 
SaSeS apcr, S/3SeS/3pcr, SyScSypcr I 
SaSpSaeo-, S/jSpS/3ecr, Sy8pSyecr ... (20) 
SaScrSaep, $/38oS/3ep, SyScrSyep j 
which may be directly verified by noting that the determinant is quadratic 
in a and vanishes when a coincides with any one of the other six vectors. 
It is easy to show by expansions like (7) that any of the two-row minors 
from the elements of the second and third columns is equal in absolute 
value to Ca, similarly for C/3 and Cy; in other words, the identities (17), 
(18), (19) are equivalent to expansion of the determinant by the elements 
of the respective rows; this is also evident from equations (17)-(19). 
Again, in (14) let r = cr, and we have 
aSaorCa + /3S/3(tC/3 4 - ySycrCy + SSScrCS + eSecrCe + pSpcCp = — <x 3 Ccr . (21) 
It is evident that by assigning various forms to the function F we may 
obtain such relations in large number. 
8. Application to the General Theory of Quadratic Vectors. 
Suppose that, in (12), F stands for any quadratic vector function, and 
let the seven vectors a, (3, ... cr be the seven axes of the function, so that 
F(a) = a 1 a, F(/3) = a 2 /3, etc. ; we have 
CL^olCol + U2/3C/3 + . . . + CL^arCcr = 0 .... (22) 
which is an illustration of the fact that the scalars a v a 2 , etc., cannot 
be assigned arbitrarily when the axes have been chosen — a noteworthy 
difference between quadratic and linear vector functions. 
Again, let cr be the variable point vector, and the other six vectors be 
any assigned constant vectors which do not lie on a quadric cone, so that 
Cor is not zero. (12) then shows that any quadratic vector F(or) is fully 
determined by its effect on any six vectors not on a quadric cone, a fact 
that might perhaps be anticipated, since the quadratic vector depends on 
eighteen scalar constants. (I have elsewhere shown that the quadratic 
vector is determined by the directions of its seven axes, aside from a term 
of the form and a multiplicative scalar constant. See Proc. Amer. 
Acad. Arts and Sci., vol. lii, No. 7 (Jan. 1917), p. 377.) 
As another illustration, take j3 = F(a) and y = F[F(a)], which we may 
abbreviate as y = F 2 (a). Let ^ = F 3 (a), etc., up to cr = F 6 (a). We then have 
by (12) an identical relation which must be satisfied by the seven vectors 
F(a), F 2 (a), . . . F 7 (a). We have evidently an analogy here with Hamilton’s 
well-known cubic satisfied by a linear vector function. Of course, to make 
