1919-20.] An Identical Relation connecting Seven Vectors. 131 
manner of writing shall be as in (3), it will be sufficient to specify the 
omitted vector. Thus (3) may be abbreviated as C(<r) or simply as Ccr. 
Similarly, if S be omitted, we may write 
CS = S . Y (Y epY a/3)Y(Y pcrY /3y)Y (Y craYye) ... (9) 
with a similar convention for the other expressions of like form. When 
necessary to avoid ambiguity, the vectors which occur may be designated 
in their order, thus : 
C3= C(cpcra/3y) . . . ■ . (10) 
or as a further illustration : 
COSpXtfTfi) = S . Y(y^pY< ? 5,T)Y(ypAVT/x)Y(yA0Y / x/3) . . (11) 
where f3, p, X, <f>, r, and p are any six vectors whatever. This may serve 
as a definition of C. 
5. Identical Relation connecting Seven Vectors. 
Let now F(a) denote any function homogeneous of the second degree 
in a in the sense that the scalar components of F(a) are quadratic 
polynomials in the components of a. F(a) may be a vector or a 
quaternion, or, of course, a scalar. Let C a, C/3, Cy, etc., be the seven 
expressions formed as in (3) or in (9) by omitting the designated vector 
from the cycle (8). The following equation must hold for all values of 
the seven vectors and for any quadratic function F : 
F(a)Ca + F(/?)C/? + . . . + F(cr)Co-= 0 . . . (12) 
The proof of this relation depends on the following : 
Lemma . — If two of the seven vectors coincide, the corresponding C’s 
become equal in absolute value but opposite in sign. 
For example, if (3 be allowed to coincide with a given a, the expression 
C/3, which does not contain f3, is unaltered, but C a, which was originally 
S . Y(V/3yVep)Y(Vy8Ypcr)V(V8eYo-P), 
now becomes 
S . Y ( V ayY cp)Y ( YySY p<r)Y ( YSeY era). 
This is the negative of C/3, for the first vector of the first group is a, 
while in forming C/3 the first vector would be y, which, counted as number 
one, makes a the sixth in order. Similarly, we may prove the lemma for 
any pair of C’s. 
It follows at once that (12) holds true when /3 is made to coincide 
with a. For Cy, C3, Ce, C p, and Co- all vanish, and the first two terms 
become equal in absolute value and opposite in sign. 
Similarly, (12) holds true when y, 8, e, p, or o- coincides with a. But 
