1892 - 93 .] Prof. Knott on Innovations in Vector Theorij. 229 
The first is simply the old thing, and the second, Ypcficr, is a very 
important quantity in the theory of the linear and vector function. 
Professor Gibbs continues : — “ It is evident that 
{px<^} . ij/==px 
{p X ^ j . a = p X [</) . a ] tfec.,” 
with other three similar equations, all of which are meant to establish 
that the associative principle holds. In quaternion notation there 
is no difficulty, the two sides of the identity in nearly every instance 
having exactly the same form. Thus, the two given above are 
Ypcfi\f/(r and Vp<^a, the o- being introduced to make the signifi- 
cance of the former at once apparent. The comment on the 
dyadic equation 
l//. {p X ^{iffXp} . cfi 
is that “the braces cannot be omitted without ambiguity.” The 
quaternion expression is i[/Y pcfm-, where there is no chance of 
ambiguity, where everything is perfectly straightforward, and 
where there is much greater compactness in form. It seems to 
me that this last equation given by Gibbs condemns his whole 
principle of notation. It shows that one grand use of connecting 
symbols is to obscure the significance of a transformation ! It is 
interesting to note — as bearing upon the intelligihility of the nota- 
tion — that Heaviside, who dotes so on the dyadic, writes <j>x p in 
the form Ycjip, so that he makes 
<f)Yp(T = — Ycrcfip ! ! 
In the course of the development of the theory of the dyadic, 
Gibbs, with his usual proneness to lexicon products, invents a few 
names (or new meanings to old ones), such as Idemfactor, Eight 
Tensor, Tonic, Cyclotonic, Shearer, and so on. These are all special 
forms of the linear and vector function ; and, excepting possibly the 
names. Professor Gibbs does not seem to have contributed anything 
of value to Hamilton’s beautiful theory. In no case, so far as I have 
been able to see, do his methods compare, for conciseness and clear- 
ness, at all favourably with Hamilton’s and Tait’s. Take by way of 
further illustration the expression of YX'p! in terms of YXpt, where 
V = cj)\ and p! = cf)p. At the very outset of the quaternion investi- 
gation we find 
YX'p — nuji'~WXp 
