1892 - 93 .] Prof. Knott on Innovations in Vector Theory. 231 
not really of any account, except in so far as it yields two very 
useful quantities, the one a vector and the other a scalar. No con- 
nection, of course, with the quaternion.” 
The quaternion, indeed, must come in, though it be but fitfully ; 
and Professor Gibbs is virtually obliged to introduce it in his treat- 
ment of the versor, which he regards as a special form of dyadic. 
To educe its properties he uses those quantities formed by a 
so-called indeterminate process from the operator He discovers, 
as any quaternionist could have told by intuition, that “ - and 
determine the versor without ambiguity.” Observe the negative sign 
before the vector ; and note the obstinacy of the writer in refusing 
to recognise explicitly the essentially quaternion character of the 
versor. 
The expression 
{2/3;8-I}.{2aa-I} 
represents in Gibbs’s notation ‘‘ a versor of which the axis is perpen- 
dicular to a and /?, and the amount of rotation twice that which would 
carry a to fB. It is evident that any versor may be thus expressed, 
and that either a or /5 may be given any direction perpendicular to 
the axis of rotation.” Here we have the quaternion idea (in one 
simple instance of it) expressed as clearly as language can express it. 
The formula just given (in which “I” represents an idemf actor ^ — that 
is, unity) transforms at once into the quaternion form ySa ( ) a/?, or 
more simply q ( ) q~^, where q is the quaternion that turns any 
vector (a) perpendicular to its axis through a definite angle. In 
this incomparable form we find for successive rotations the equation 
rq{ )q-h'-^ = rq{ )(r^)"i 
Taking 0^^q = Yq, 6^r = Vr, 
we have 
V(n^) _ V.VrVg -1- -f 
S VrV^ -1- 
a perfectly general formula which Gibbs takes half a page to 
demonstrate, and even then his demonstration applies to versors 
only. 
