98 
Proceedings of Royal Society of Edinburgh. [sess. 
Proposed Extension of the Powers of Quaternion Differ- 
entiation. By Alexander M‘Aulay, Ormond College, 
Melbourne. Communicated by Professor Tait. 
(Read December 15, 1890.) 
It will, I think, be acknowledged that Quaternions, while pro- 
viding for the physicist a machinery much more natural and 
graceful than the Cartesian, for all conceptions strictly geometrical, 
do not at present afford equal facilities for the consideration of 
questions involving differentiation. It is true that there is one 
well-known symbol of differentiation of great utility, which enables 
Quaternions to deal in a suitable manner with many such questions ; 
but there are left whole classes of differentiations in which the 
symbol is of no avail. 
This has led me to the consideration of other symbols of differen- 
tiation, and to a slight generalisation of the powers of the symbol 
already mentioned. I had thought it necessary only to define the 
extensions here referred to, and proceed to apply them. But Pro- 
fessor Tait, while kindly procuring me this opportunity of bringing 
forward my views, has given me fair warning that the repugnance 
of physicists to some of my notation may prove an insuperable 
obstacle to their paying any attention to investigations conducted 
in that notation. This personal reference will explain why, in the 
present short paper, an apology for, and explanation of, the methods 
are entered into, much more detailed than could otherwise be con- 
sidered advisable or even justifiable. 
In what follows, after an explanation of the proposed changes of, 
and additions to, quaternion differential notation, a brief account in 
the abstract is given of the reasons for and against what can be 
called innovation; and the rest of the paper is devoted to some 
examples of the application to the theories of elasticity and electro- 
statics. I may here remark that, in a short paper like the present, 
it is impossible to do full justice to the views enunciated, because 
for this purpose it would be necessary to go over a large part of 
the ground covered by mathematical physics : but if the slight 
variations from previous custom indicated below do not meet with 
