1919-20.] An Identical Relation connecting Seven Vectors. 137 
where the first vector in brackets denotes the omitted vector, and the 
second denotes the one which was acted on by v. Similarly, if v acts on <j, 
V 2 Cp = H[p<r] . ...... (39) 
and it is evident in virtue of the lemma above that 
H[a-p]=-H[pcr] (40) 
Where needed to avoid ambiguity, we may designate the vectors which 
occur in the result, thus : 
R(a(3ySe) = 2Sa/?eSySeS . Vay Yf38 - 2Say e S/?S € S . YajSV yS . . (41) 
which may serve as definition of the function H of five vectors. The 
factor 2 is retained for convenience. 
Returning now to our general identity (12) and operating with v 2 , 
o* being the operand, we obtain 
F(a)H[a<r] + F(/3)H[j8<r] + . . . + F(p)H[p<r] + C<rV 2 F(o-) = 0 . (42) 
which expresses v 2 F(a-) in terms of F(a), F (/3), etc. It is clear that this 
is an identity of the same sort as the familiar 
V<£/3 = XfJLffiV + fJLVcf>\ + 
where 0 is a linear vector function and v acts on p : the effect of v on the 
function is expressed in terms of functional operations. 
By giving to F various values we obtain numerous relations connecting 
the H’s. Thus, if F(«) = ct 2 we have 
a 2 H[acr] + /3 2 H[/5cr] + . . . + p 2 H[p<r] + 6Co = 0 . . (43) 
which expresses one of the C’s in terms of six ITs. We note that this 
identity involves only six vectors. 
10. Identities obtained by replacing Vectors by v. 
We come now to the third of the three methods mentioned in Art. 6, 
namely, replacing one or more vectors by the operator v. Since v is a 
vector, we may, in any of the foregoing identities, let any one of the 
vectors present become v and we shall have an operator which vanishes 
identically, and which, when applied to any expression whatever, yields 
an identically vanishing result. 
To illustrate, let F(«) = aSaT and use (42). We have first 
aSarH[arr] + /3S/?rH[/2cr] + . . . + pSjorH[pcr] + C<tV 2 (o-Sctt) = 0 . (44) 
where v in the last term acts on or. Now 
SA.V(<tS<tt) = — /YS<xt — <xS/Vt 
(45) 
