1892 - 93 .] Prof. Knott on Innovations in Vector Theory. 219 
question to write Va = y/5~^ or Sa = «/5"f Such transformations 
are meaningless."^ The point I wish to emphasise is that Hamilton’s 
notation does not even suggest the possibility of such a transforma- 
tion. On the other hand, Grassmann’s, O’Brien’s, Gibbs’s, and to a 
certain extent Heaviside’s, notations do suggest such a possibility. 
It is certainly inexpedient, to say the least, to use a notation strongly 
resembling that for the multiplication of ordinary algebraic quantities, 
but having no corresponding process by which either factor can be 
carried over as a generalised divisor to the other side of an equation. 
Heaviside uses af^ in the sense of - Sa^, and to this notation 
exactly the same objection applies. 
It is possible that Hamilton’s notation might be improved, but 
certainly not in the way advocated by Gibbs. So long as we deal with 
two vectors only, there is no doubt a slight saving in symbols by use 
of forms like a(3 or a./? instead of - SayS. But a x y8 is no more easily 
written than is Yaj^. Hamilton’s form SaySy is as easily written 
as Heaviside’s aV f3y, besides being more symmetrical and expressive ; 
while a.fS X y is clearly not in the running for compactness, per- 
spicuity, or symmetry. Again, such a form as VaySSyS becomes 
with Gibbs a x (3(y.S), while Heaviside would probably write it 
YajS.yS. The quantity aS^yS would take the forms a{/3.y x 8) 
and a.ySVyS. Gibbs cannot write Va/3y, which must be put into 
the form - a{/3.y) -i- a x (/5 x y), a hideous parody of aSySy YaV J3y. 
The peculiar perspicuity of Hamilton’s notation arises from the 
fact that the S and V are thrown out in such relief from amongst 
the Greek letters used for vectors and the small Ptoman letters used 
for quaternions and scalars. A glance suffices to tell whether a 
quantity is scalar or vector. We know at once what kind of 
quantity we have to deal with before we are called upon to inquire 
into its composition. Herein lies one great merit of the prefix 
method in such a calculus. Heaviside, to a large extent, destroys 
the contrast between the quantities and the selective symbols by 
using capital letters for vectors. In print the vectors are made 
heavy, and of course stand out prominently enough. But a vector 
analysis is a thing to he used ; and it is hopeless with pencil or pen or 
chalk on a blackboard to try to prevent confusion between A and A. 
Heaviside virtually condemns his whole system by suggesting a 
Thus we cannot transform Sin30 = « into Sin0= 
