224 
Proceedings of Royal Society of Edinhurgli, [sess. 
bits which are either scalar or vector, and defining them inde- 
pendently of one another.* Thus V.Vw is defined to have the 
meaning 
and a parenthetic note is added to the eflect “that no meaning has 
been attributed to V before a vector.” One would think that the 
most natural inquiry for a vector analyst to make after he had 
formed the conception of the quantity would be — “What Avill 
be the effect of V on a vector?” But no, says Gibbs in effect, be 
not so hasty, or you may get that “ trivial ” thing, a quaternion. So, 
in heginning the study of these functions, he makes no less than 
four definitions, the definitions, namely, of Vw, - SVw, VVw, and 
- V ^co, — the meanings of all of which follow in quaternions from 
one definition. The whole principle of Gibbs’s and Heaviside’s 
methods is, in fact, to represent more or less concisely certain 
quantities which occur frequently in mathematical physics ; and, 
in doing so, they simply adopt from quaternions as much as they 
think they need. Unfortunately for themselves they take the shell 
and throw away the kernel. 
For even with these four definitions — and others are introduced 
later as the subject is developed — Gibbs finds his system lacking in 
flexibility and true vitality. It has no power of self-growth, but 
has to be fitted with here an arm and there a leg, like a mechanical 
puppet. Then he has, so to speak, to lubricate its joints by pouring 
in the definitions of four other functions, with as many new symbols. 
One of these is the Potential — the others are called the Newtonian, 
Laplacian, and Maxwellian. They are symbolised thus — Pot^ New, 
Lap, Max. 
The meanings of these functions will be at once evident when 
they are exhibited in quaternion form. Thus, as is well known. 
from which at once 
* The query suggests itself, could these bits have been discovered without 
quaternions as a guide ? 
( d^ d^ dL\ 
\dP dy‘^ dz^J ’ 
