1919-20.] An Identical Eelation connecting Seven Vectors. 139 
all linear in p, while each is of the fourth degree in one vector and 
quadratic in the rest. The identity as a whole is thus of the fourth 
degree in a, /3, y, S, e, and of the third degree in p. Expressions like the 
L’s of (53) present themselves at once when we attempt to solve the 
equation WFp = 0, where Fp is a quadratic vector function of p to be 
specified by its axes. The details of this question must be reserved for 
a future work. 
11. Extension of the Method. 
In the foregoing paper our results have been obtained from the 
identity (12) connecting an arbitrary quadratic function, of seven different 
vectors successively, with scalars which vanish when six of these vectors 
lie on a quadric cone. 
If we can form scalar functions of ten vectors which vanish when these 
lie on a cubic cone, we may evidently extend our results to an arbitrary 
function of the third decree, and so on. 
In particular, if for the vectors in question we choose F(«), F 2 (a), F 3 (a), 
etc., we shall have an identity satisfied by the functional operation of 
whatever degree. How far the coefficients are invariant under varying a 
appears a question of interest. The form of the coefficients may be 
determined by the methods of Art. 7. Taking the cubic case as an illus- 
tration, assume that any function F homogeneous of degree three in one 
vector satisfies an identity of the form 
F(ai)Ci + F(a 2 )C 2 + . . . +F(a n )C n = 0 . . . (54) 
where the a’s are any eleven vectors whatever and the C’s are to be 
determined. Let F(aj) = cqSoqrSa-LTr, where t and 7r are any two vectors. 
The identity becomes 
ajSajrSajTrC]^ + a 2 Sa 2 rSa 2 7rC 2 + . . . + a 11 Sa 11 rSa 11 7r = 0 . . (55) 
Let 7r = Va 10 a 1;l and T = Va 8 a 9 , also multiply into Va 6 a 7 and take scalars, 
giving the five-term identity 
Sa 1 a 6 a 7 Sa 1 a s a 9 Sa 1 a 10 a 11 C 1 + . . . + S V*6 a 7 Sa 5 V9 Sa 5 a lo a lA5 = 0 . (56) 
We may at once write down as many equations of this form as there 
are ways of pairing off the six vectors a 6 , a 7 , . . . a n . Thus the first five 
C’s can be found, and similarly the others. To prove that the C’s are 
uniquely determined in this manner would too far lengthen the discussion. 
It is probable that the C’s in the third and higher degrees may be 
much more neatly expressed by products analogous with Hamilton’s 
expression (3). 
{Issued separately October 13, 1920.) 
