136 Proceedings of the Royal Society of Edinburgh. [Sess. 
Next let <j coincide with a, giving 
S/3rL(ySep[V/3a, a]a) + SyrL(Sep[Vya, a]a/3) + . . . + SprL([Ypa, a]a/3ySe) = 0 (32) 
for the last term of (31) vanishes when a coincides with a. Finally, let 
t have the value Yep, and we have the following relation which connects 
three L’s : 
S/?e/>L(ySep[V /3a, a]a) + SyepL(Sep[Vya, a]a/3 ) + SSepL(ep[VSa, a]a/3y ) — 0 (33) 
To interpret the meaning of these three L’s, we note that they are 
derivable from the expression Co- by regarding /3, y, and S, respectively, as 
variable. Thus, for example, 
L(y8ep[V/3a, a]a) = — L(a[V/3a, a]ySep) 
= — S/3aV . SaV . Ccr . . . (34) 
where both v’s act on /3 only. The vanishing of this expression would 
imply that the polar plane of the vector V 6a with respect to the cone 
Co- = 0 (where /3 is the variable point vector) passes through the vector a. 
It is probable, however, that the most interesting results come from 
the use of the operator v 2 . The expressions obtained by acting with v 2 
on one of the C’s have remarkable properties. The easiest way to operate 
with v 2 is, usually, to act first with S\v, then write \ = v and operate 
again. Thus, if v acts on p we have 
SA-V(Sp-pSvp) = — SpASvp — S/xpSi/A. .... (35) 
whence writing v for X we have 
V 2 (S/xpSi/p) = 2Sp,v ..... (36) 
With this result in mind, we see by inspection of (7) that 
V 2 Ccr= 2Sya/?SSe/3SVySVae — 2SyS/3Sa.e/?SVyaVSe . . (37) 
Since the original expression Co- was proved to be unaltered in absolute 
value by interchanging any pair of vectors, the same must hold for the 
right of (37), and the rules for changes of sign must also hold. It follows 
that the expression can be expanded in a great variety of ways, according 
to the original order of writing in expressions like (3). The reader may 
be interested to examine what relation, geometrically, holds among the 
five vectors which are present, when the right of (37) vanishes. 
If we operate again with v 2 , letting v‘ act this time on /3, the result is 
zero; that is, the right of (37) is a harmonic function of /3. Whence it 
is also a harmonic function of a, y, S, and e. This suggests the notation 
(when v acts on p) 
V 2 Ccr — H[crp] . 
(38) 
