134 Proceedings of the Royal Society of Edinburgh. [Sess. 
the analogy complete it must he shown that the coefficients Ca . . . Ccr 
become invariants (or covariants) when the vectors (3, y, ... cr are chosen 
as above. The investigation of this interesting question must be reserved 
for a separate paper. 
9. Identities obtained from (12) by Differential Operators. 
An almost unlimited number of identities may be obtained from (12) 
by first multiplying through by any function of v and then allowing v 
to act on one of the vectors, as <r, or on the constituents of F. 
For example, multiply into v> take scalars, and allow v to act on < 7 . 
We have, if F does not contain 0 - implicitly, 
SF(a)V . Ca + SF(/?)V • C/3 + . . . +SF(p)V.Cp + C«rSVF(cr) = 0 . (23) 
As a special case, let F(a) = aSar as in (14), when (23) gives 
SarSaV . Ca + S/3 tS/ 3V . C/3 + . . . + S/orSpV . C/o — 4 CctSo-t = 0 . (24) 
where v is understood to act on <r alone. 
The expressions SaV • Ca, S/3v . C/3, etc., which occur in this result 
deserve a moment’s special consideration. Since Ca is a homogeneous 
quadric function of or whose vanishing, when a- is the variable point vector, 
defines a quadric cone, it is clear that SaV • Ca defines, by its vanishing, 
the polar plane of a with respect to this cone. The importance of such 
polar relations is too well known to require emphasis here. It is evident 
that, by acting on the various C’s with operators of the form SXv and 
letting v act on any one of the vectors present, we may obtain a very 
great number of such polar planes. A systematic notation is desirable. 
It is apparently sufficient to designate, in order, the omitted vector, the 
vector whose polar is being taken, and the operand. Thus we may write 
SaV . Ca = — L[a, acr ] when v acts on < 7 . As a further illustration, if we 
start with the expression (3), act on it with SXv, and regard p as the 
variable point vector, we may write 
SXV. Co- = -L[<r,Ap] (25) 
where we understand : 
1. The first vector in brackets denotes the omitted vector. 
2. The second vector occurs in the operator, i.e. we are finding the 
polar of this vector. 
3. The last vector in brackets is the operand or variable. The letter 
L may remind us the expression is linear in the vectors X and p. It is 
symmetrical in these vectors. 
This notation is doubtless far from ideal, but is compact, and sufficiently 
specific for our present purpose. When necessary to prevent ambiguity, 
