1892-93.] Prof. Knott on Innovations in Vector Theory. 223 
the suggestion. Its square is minus the well-known Laplacean 
operator 
0 , 2 + 0 ^ 2 + 0^2 
or I +1-7- 
dx) 
This minus sign is, of course, objected to by Heaviside and Mac- 
farlane, who get rid of it (and a good deal else beside) by their 
assumption that the square of a unit vector is one. 
For convenience of reference I shall give a table comparing the 
V notations of the innovators with the real quaternion quantities. 
u is any scalar quantity, w any vector quantity. 
Hamilton-Tait. 
Gibbs. 
Heaviside. 
Macfarlane. 
V u 
V u 
V u 
V (jO 
non-existent in 
any. 
VV(0 
V X oj 
W (O 
Sin V w 
SV(o 
— V . O) 
- Vco 
- cos V (O 
(N 
l> 
- V . V w 
- Vhi 
t> 
1 
V2(0 
— V . V o) 
- 
3 
L> 
1 
VSVw 
— V V .(0 
- V ( V (o) 
- Sin V (cos V w) 
VYVw 
V X V X (o 
VVVVco 
Sin V (Sin V w) 
V-'o) 
non-existent in any. 
and so on indefinitely. 
Macfarlane has (and Heaviside should have) a quaternion-like 
quantity Vw, which is expressible in the quaternion form - KVw. 
Again, in this system, there is (or should be) a quantity V (Vw) which 
is not the same as — the result, of course, of the non-associative 
property of their vector products. Its value in quaternion sym- 
bolism is - VKVw or V2(o-2VSVco, which Mr M'Aulay might 
express in the form - V w, V,. 
Consistent (so far) in his rejection of anything that suggests a 
(quaternion product, Gibbs contents himself by picking out those 
word, which does not euphoniously combine with other terms ; to use Delta 
is, of course, out of the (question. Nahla is certainly euphonious, as any who 
have used it in lecturing can testify. Being quite innocent of any previous 
scientific significance, it has just the meaning that is q)ut into it. Nahla 
occurs only in quaternion analysis; for the operator is not used by Gibbs in 
its full sense, and that Avhich Heaviside and Macfarlane represent by v is not 
Nabla. 
