1892 - 93 .] Prof. Knott on Innovations in Vector Theory. 213 
vectors, which he symbolises by the notations a.(^ and ax/?. The 
product a X /? he calls the longitudinal translation of a along /? ; and 
the product a.yS the lateral translation. The geometric meanings of 
these are sufficiently obvious. Thus the translation of OA or a 
along OB or has, so to speak, a longitudinal part and a lateral 
part. The longitudinal translation (ax /3) is represented by the 
translation of OM (the resolved part of a along from 0 to B. 
It is measured by the product OM.OB or a?mosAOB, where a h 
are the lengths of the vectors a /3. Similarly, the lateral translation 
{a.j3) is represented by the translation of OK (the component of a 
perpendicular to fS) from O to B. It is measured by the area of 
the parallelogram contained by a and Of course, these are simply 
Orassmann’s “ inner ” and “outer” products. It must be noted, how- 
ever, that O’Brien’s a./? is not the same thing as Hamilton’s Va/?. 
To represent this conception he introduces the Directrix, and uses 
for it the symbol D. Thus D {a./3) is a directed line drawn per- 
pendicular to the plane containing a/3, and of a length numerically 
equal to the area of the parallelogram a./?. He then uses Da, D/?, 
Dy, somewhat in the sense of Hamilton’s i,j, k. In an interesting- 
footnote he points out very precisely the difference between his 
system and Hamilton’s, noting, for example, that the latter identifies 
a and Da, and finds in consequence that the square of any unit 
vector is equal to negative unity. He sees very clearly that unit 
vectors must have their squares all equal, but admits that his own 
value of positive unity is only an assumption. He concludes 
with these words : “ If in any way I could show that - I was the 
proper value for a unit of longitudinal translation, I should have 
aa = a X a-P a.a= - 1.” O’Brien applies his methods to general 
dynamics, and gives, besides, some applications of the linear and 
vector function in one of its standard forms, and also of the operator 
a0]^ + /?02 + y9g. In one of his early papers Hamilton refers to 
O’Brien’s work in terms of high praise. It will be seen that 
O’Brien’s methods of establishing his calculus are identical in 
principle with the long subsequent ones of Gibbs and Heaviside. 
(2.) Professor Gibbs’s position is best described in his own words 
{Nature, vol. xliii. pp. 511-12) : — 
“The question arises, whether the quaternionic product can claim 
a prominent and fundamental place in a system of vector analysis. 
