138 Proceedings of the Royal Society of Edinburgh. [Sess. 
so that writing v for X we have 
V 2 (<rSo-r) - — VScrr — StV . <r 
= 2 T (46) 
whence (44) becomes 
aSarH[a<x] + /3S/IrH[/3o-] 4- . . . + /oSprH[po-] + 2rCcr = 0 . (17) 
which so far is merely an identity connecting the six vectors of the 
cycle (4) with the vector r which occurs in a linear manner. In other 
words, this identity is of the same sort as (14), but simpler in that <r no 
longer enters, and the H’s involve only five vectors. If we now write 
V for r we have a relation expressing v in terms of the six operations 
SaV, S/3v, • • • SpV, where the six vectors a, /3, ... p are any vectors 
whatever, namely, 
H[acr]aSaV . + H[y8o-]/3S/3v . + . . . + H[/oo-]pSpV . = - 2 C<t . V . (48) 
If we multiply into v we have v 2 expressed in terms of the squares of 
the same operators, thus : 
H[ao-]S 2 aV. + H[/lo-]S 2 /?V . + . . . + H[po-]S 2 pV . = - 2Ccr . V 2 . (49) 
As a final illustration I shall derive an identity which I have found 
useful in working with quadratic vectors. Beginning with (47), multiply 
through with pr and take scalars. The last two terms disappear, giving 
the following five-term identity : 
SaprSarH[a(r] 4- S/DprS/^rH^cr] + . . . + ScprSerHfeo-] = 0 . (50) 
Next write v for r, and we have the identically vanishing operator 
SapV . SaV . H[ao-] + S/?pV . S/3V . H[/kr] + . . . + SepV . SeV . H[ecr] = 0 (51) 
which involves only the six vectors of the cycle (4). Let the left side 
of (51) operate on our original expression (3), p being taken as the 
variable, that is, p is acted on by v wherever it occurs in (3) but not 
where it occurs in the operator itself. Note that by our definition of the 
function L w T e have 
Sa pV . SaV . Ccr = L(a/3ySe[Vap , a]) . . . . (52) 
with similar expressions for the other terms of the result. We then have 
when (51) is multiplied into (3) 
H[ao-] . L(a/3y8e[Vap, a]) + H [/kr] . L(a/3y8e[V/lp, /?]) + . . . 
+ [Hecr] . L(a/?y8e[Vep, *]) = 0 . . . (53) 
This is a scalar identity involving the six vectors of (4), for the H’s are 
specified, for brevity, by writing the vectors which do not occur. (We 
obtained H[acr] from Ca by letting v act twice on cr.) The five L’s which 
occur in this identity are of the same character as those in (33), but differ 
in that they are all obtained from Co- by using p as operand. They are 
