218 Proceedings of Royal Society of Ecl'Miorcjh. [sess, 
is far from simple.” I fancy Gibbs has in mind the definition of a 
quaternion as the sum of a scalar and vector, although he nowhere 
tells us explicitly what he imagines a quaternion fundamentally to 
be. In like fashion Heaviside writes : “ The quaternion is regarded 
as a complex of scalar and vector.” The pure analyst may so regard 
it ; but to the physicist it is made up of tensor and versor. Its 
property of being decomposable into scalar and vector parts, each 
with a geometric meaning at first sight distinct from its own funda- 
mental characteristic, is an absolutely invaluable one. The quater- 
nion comprehends within itself the conceptions of a rotation, a stretch- 
ing, a vector area, and a projection. You may extract whichever 
part or parts may serve your purpose for the moment — they are all 
there uniquely determined when the quaternion is given. Here, 
truly, is a king of quantities. ‘‘Upon earth there is not his like.” 
(5.) In considering the claims of the rival systems of vector 
analysis, we will first glance at the notations suggested. 
A good notation for a new calculus is half the battle won; and a 
notation must, of course, be in harmony with the principles of the 
calculus. Having to his own satisfaction abolished the quaternion. 
Professor Gibbs proceeds to argue that “we obtain the ne plus ultra 
of simplicity and convenience if we express the two functions [Yaf3 
and - Sa^] by uniting the vectors in each case with a sign sugges- 
tive of multiplication.” Consequently he represents these functions 
in the forms ax ^ and a./S. These are the forms used long ago by 
O’Brien, only he used them the other way about. 
How there is a serious objection at the very outset to such a form 
as a X ^ for the vector product of a and /?. There corresponds to 
it no quotient amenable to symbolic treatment. The reason, of 
course, is that a x /3 is not a complete product. It is only a part of 
the complete product, which Hamilton writes aj3. Given the 
quaternion equation a/3 = q, any one quantity is uniquely determined 
if the other two are given. The quotient a = q/jS has an intelligible 
meaning. But it is impossible, in spite of the suggestiveness of the 
form, to throw Gibbs’s ax p = y into any such shape as a = y~ 
How, in the quaternion notation, we have Yaf — y, Sa/5 = a, where 
the selective symbols V and S pick out two important parts of the 
complete quaternion j^roduct. Since the essence of a selective 
symbol is the partialising of the conception, it is evidently out of the 
