220 Proceedings of Royal Society of Eclinhiirgh. [sess. 
suffix notation for manuscript ; the result being that vectors and 
scalars are distinguishable only on close inspection. The conditions 
for a good notation are: — (1) an unmistakable difference between 
easily written symbols for scalar and vector quantities ; (2) the 
scalar and vector parts of products and quotients thrown out in 
clear relief. This second is quite as important as the first condition. 
It is evident that, so far, Hamilton’s notation easily holds its own. 
It is easy to see that Professor Gibbs is compelled, for mere con- 
sistency’s sake, to object to the selective principle of notation. He 
refuses to recognise that the scalar and vector products are parts of 
a complete product. The one he calls the direct product — an 
atrocious misnomer — and the other the shew product ; the idea 
being, I suppose, that this product exists only when the vectors are 
inclined or skew towards one another. Presumably the recognised 
term vector product smacked too much of Hamilton and his ways, 
although there is no doubt it is infinitely more appropriate, even 
from Professor Gibbs’s limited point of view. 
(6.) One of the most important simjDlifications in quaternions is 
the identification of vectors with quadrantal quaternions, or of unit 
vectors vdth quadrantal versors. 
Beginning with the quaternion quotient q — aj/3, we are quickly 
led by space considerations to study those quaternions which rotate 
a given vector through a right angle. Now, suppose we have two 
quadrantal quaternions I and that we operate on the vector a 
which is perpendicular to the axes of both, then it is easy to show 
that 
la -}- 1' a = (/ + I')a 
gives a quadrantal quaternion (/+/') bearing to / and / exactly 
the same relation which would exist were I and I' vectors. That 
IS, quadrantal quaternions are added and subtracted according to 
the recognised rules for vector addition and subtraction, which so 
far, be it noted, are all we know about vectors. Is there any d 
priori reason why a vector, acting on another vector at right angles, 
should not be regarded as a versor ? Not only is there no reason 
against such a conception, but every vector analyst, excepting, per- 
haps, O’Brien, has made, and does make, that very assumption. 
As we said above (§ 3), Gibbs states it explicitly. That much 
being admitted, and the associative law being assumed to hold in 
