164 Proceedings of Royal Society of Edinburgh. [sess. 
versus Coordinates .” In the Phil. Mag. (June 1885) he has a 
paper On the Electro-magnetic Wave-Surface , from which I quote 
as follows : — 
“ Owing to the extraordinary complexity of the investigation 
when written out in Cartesian form (which I began doing, hut 
gave up aghast) some abbreviated method of expression becomes 
desirable, I may also add, nearly indispensable, owing to the great 
difficulty in making out the meaning and mutual connections of 
very complex formulae. In fact, the transition from the velocity- 
equation to the wave-surface by proper elimination would, I think, 
baffle any ordinary algebraist, unassisted by some higher method, 
or at any rate by some kind of short-hand algebra.” 
Dr Heaviside then considers the fitness of quaternions to supply 
the want, and refers unfavourably to my treatment of Fresnehs 
wave-surface. (Hamilton, in his Elements , highly commended even 
the first (very imperfect) form of it !) Next, Dr Heaviside de- 
velops a vector-system of his own, and with considerable labour 
and at comparatively great length obtains results which (as rendered 
for me, from his Yolapuk into Quaternions, by Dr Knott) are 
found to be precisely what Quaternions directly and naturally give 
in a few lines ! Dr Heaviside, in a semi-defiant attitude, gives 
away his case by asserting that his process “ cannot be made more 
direct, or shorter, except of course by omission of steps, which is 
not a real shortening Hence, and also of course, the Quaternion 
wins, hands down, against his new system. This particular 
application (including the simple proof of the identity of the various 
forms of the equations, which seems to have given Dr Heaviside 
very serious trouble) was made (partly by Hamilton, partly by 
myself) in 1859, and is to be found in §§ 382, 385 of the first 
edition of my Quaternions , or §§ 182, 439 of the third edition. 
The change of a few letters, to introduce the slightly wider data 
of the new problem, is all that is required.” 
