236 Proceedings of Royal Society of EdAnhurgli. [sess, 
elude that Gibbs, after all, does use the expressions c?cra>, dpm, 
and V(o in the sense of quantities. To what end, then, was his 
Nature letter written, and why does his pamphlet not contain the 
explicit definitions of the meanings of such quaternions ? 
A favourite argument used alike by Gibbs and Heaviside is, that 
even avowed quaternionists work mostly with vectors and scalars 
and rarely with quaternions. But this surely betokens a total 
misapprehension of the whole significance of the calculus. In 
trigonometrical analysis we work mostly with sines, cosines, tangents, 
secants, and so on. But, because sin 6 occurs a myriad times for 
one occurrence of 6 itself, do we not then deal with angles or arcs in 
trigonometry? By his notation Gibbs tries — with what success we 
have seen — to sink the quaternion quite out of sight. But Heavi- 
side uses Hamilton’s vector notation ; and this notation emphasises 
the truth, that a quaternion is being dealt with. The selective 
symbols V and S are as quaternionic as T, U and K. The equation 
Ypcfxj = 0 
is as fundamentally a quaternion equation as is 
V.q6q-^ = 0 
And, although Gibbs gets over a good deal of ground without the 
explicit recognition of the complete product, which is the difference 
of his “ skew ” and “ direct ” products, yet even he recognises in plain 
language the versorial character of a vector, brings in the quaternion 
whose vector is the difference of a linear vector function and its 
conjugate, and does not hesitate to use the accursed thing itself in 
certain line, surface, and volume integrals. 
The principal arguments of the present paper may be conveniently 
summarised thus : — 
(1.) The quaternion is as fundamental a geometrical conception 
as any that Professor Gibbs has named. 
(2.) In every vector analysis so far developed, the versorial 
character of vectors in product combinations cannot be got 
rid of. 
(3.) This being so, it follows as a natural consequence that the 
square of a unit vector is equal to negative unity. 
(4.) The assumptio7i that the square of a unit vector is positive 
unity leads to an algebra whose characteristic quantities 
