1892 - 93 .] Prof. Knotfc on Innovations in Vector Theory. 227 
analysis, have exactly the properties that belong to other similar 
operators, and consequently are amenable to analytical transforma- 
tions of a general kind. The operator v? with its wondrous poten- 
tiality of transformation and of meaning, was regarded by Hamilton 
himself as one of his most remarkable creations. We have as yet 
only a glimmering of its full significance. As if to retard the fuller 
revelation, we are now asked to hack it into pieces small — into 
Curl, Div, Lap, Max, Hew, and Pot. 
In the table of comparisons given above it was noted that Vw 
had no place in the vector analysis of Professor Gibbs. Later 
on, however, he introduces a function resembling Vw in appearance. 
He calls it the “dyadic,” which is a vector operator (see below) but 
in this case “ represents the nine differential coefficients of the three 
components of [the vector w] with respect to x, y, and just as the 
vector Vu represents the three differential coefficients of the scalar 
u with respect to x, y, and 2 :.” 
That an operator can represent nine quantities just as a vector 
rej^resents three is a novel mathematical truth ; but it is difficult to 
see what is gained by this dyadic definition of wffiat ought to be a 
quaternion. On inspection we find that Gibbs’s functional operator 
Vo) corresponds to the quantity So-V.w with the o- left out. In other 
words, it is what might be regarded as the operator on cr if the ex- 
pression S ( )V.w could be contemplated from that point of view. 
(8.) It seems expedient to look somewhat closely into Gibbs’s 
system of dyadics, which Heaviside regards with such high admira- 
tion. For present purposes it will be necessary, occasionally, to use 
Gibbs’s notation ; but I shall represent his quantities and operators 
also in the quaternion symbolism. As usual, he starts off with some 
four or five “definitions,” which of itself suffices to show how lacking 
in the elements of self-growth his system is. 
If we form the quantity aSA.y, we may consider it as an operator 
aSA. acting on p. Professor Gibbs’s notation for the quantity is 
- aA.p ; and then the expression aA he takes as his symbol of opera- 
tion. That is, he uses the recognised symbolism for a product to 
represent an operation which by itself has no existence, and which 
possesses in its distributive quality only one of the ordinary attri- 
butes of a product. This operator he calls a dyad. The operator, 
aA -f ySft -f yVf 
