1890-91.] Mr A. M'Aulay on Quaternion Differentiation. 
99 
that studied discountenance from all authorities in the matter, which 
I have been assured they must, I could give in a future paper some 
few applications of the methods which would probably prove in- 
teresting to physicists, and which cannot be treated readily, if at all, 
by methods other than those in question. 
As is well known, Hamilton’s symbol V may be defined by the 
equation 
,d .d 7 d 
v = l J x +J Ty + k dz 
where i,j, k, x , y , z have the usual meanings. Hamilton himself 
did not examine the utility of V . This explains, most likely, why 
he did not state more exactly how to consider to what symbols the 
differentiations implied should refer. In the simplest applications of 
V these differentiations will refer to that symbol, and only to it, 
which immediately follows the operator. Very little practice in the 
application of V to physical questions serves to show that, if we are 
to be bound by this rule, at least seven-eighths of the potential 
utility of V must be sacrificed. Professor Tait has recognised this 
in the 3rd edition of his Quaternions , where he freely separates the 
V from the symbol or symbols it affects ; and, according to a well- 
known custom, indicates the connection between the operator and 
the symbols affected by attaching the same suffix to both. This I 
had already done in a paper to be referred to immediately. We 
seem then to be led by a natural process to the following state- 
ment : — 
The operator V and its operand or operands may have any relative 
positions in a term which are convenient , the connection between them 
being indicated in the usual way by suffixes. 
For the propriety of this view, I now wish to contend, although 
at the time of writing the 3rd ed. of Quaternions Professor Tait 
was not prepared to endorse it in full. Notwithstanding that it is 
immaterial how great, in a term, be the separation of a V from the 
symbol affected, so long as the V is on the left of the symbol, he* 
emphatically — observe his italics — lays down the law, that the V 
must not be removed to the right of the symbol, even the immediate 
* With regard to the use of v Professor Tait ( Quaternions , 3rd ed., § 149) 
says: — “The precautions necessary in such matters are twofold — ( a ) The 
operator must never be placed after the operand ; ( b ) its commutative or non - 
commutative character must be carefully kept in view. ” 
