1890-91.] Mr A. M'Aulay on Quaternion Differentiation. 101 
Now, what possible objection can there be to writing this last 
equation in the form 
( 1 ). 
On the left of this equation the V , as indicated by the suffixes, refers 
to the symbol immediately preceding it. On the right it refers to 
both the symbol immediately preceding and also to that immediately 
following. Certainly the expression on the right can be written 
V5)V@ + V<gV2); 
but in this shape we at once lose the connection of form between 
the vector in question and the corresponding linear vector function 
V5)( )<g. 
Professor Tait suggests in his treatise how such vectors as V l 
above may be treated. He shows how we may prove properties of 
the vector, and even suggests a method of writing down the vector. 
Adopting that suggestion, we should have, instead of equation 
(1) above 
- S V A.</>p = -SVA.V * 
To explain my notation I will quote from a paper f on this sub- 
ject, written in 1884 : — 
“ If <f> be any linear quaternion function , <£ itself being a function 
of the position of a point 
. _dcf>i + d<f>j ^ defile 
dx dy dz 
It is necessary, as will be seen below, that this A should be dis- 
tinguished from V . 
Whenever numerical suffixes occur in this paper it will be to 
* Tait’s Quaternions, 3rd ed., § 508. The following words of Professor Tait 
seem to me to form a powerful argument in favour of my natural notation : — 
“ The highest art is the absence of artifice. .... The difficulties of Physics 
are sufficiently great in themselves to tax the highest resources of the human 
intellect ; to mix them up with avoidable mathematical difficulties is unreason 
little short of crime In Quaternions, a subject uniquely adapted to 
Euclidian space, this entire freedom from artifice and its inevitable compli- 
cations is the chief feature What is required for Physics is, that we 
should be enabled at every step to feel instinctively what we are doing. Till 
we have banished artifice we are not entitled to hope for full success in such 
an undertaking” (Tait, “On the Importance of Quaternions in Physics,” 
Phil. Mag., 5th series, vol. xxix. pp. 84-97). 
t Mess, of Math. , vol. xiv. p. 26. I have substituted in the present paper 
A for the v' of the original paper. 
