“2 480 
NATURE 
[SEPTEMBER II, 1902 
vector should be denoted by 7x+7y+4z. As in the theory of 
he complex variable in two dimensions the result of any number 
of successive operations always preserved the fundamental type 
a+-26, so it was desirable that the result of the successive opera- 
tions of his vectors should issue in an equally simple fundamental 
type. This end he found he could attain if he discarded the 
commutative principle which hitherto had barred his own pro- 
gress and that of others, yet preserving the distributive and 
associative principles, and finally one happy evening he arrived 
at the beautifully simple laws by which the symbols of this 
Calculus act upon each other ; that not only 7?=/?=/"= —1, 
but also that 7= —ji=h, fkR= —hj=i, kAt= —th=7. 
Though it was thus—as the product, that is, of two vectors — 
that the Quaternion first presented itself to Hamilton, he of 
course saw that it immediately followed that it might be re- 
garded as the ratio of two vectors, in other words the operation 
which turned one vector into another. In fact in the more syn 
thetic exposition which is contained in ‘‘The Elements” he 
makes this latter the starting definition of the Quaternion. 
It is noteworthy that this, the more complete and systematic 
presentation of the subject by its illustrious author, may be said 
to owe its origin to the keen interest my predecessor, Prof. 
~ Tait, took in the new Calculus, of which, as you know, he ever 
afterwards remained the most ardent champion. This interest 
led him to seek from Dr. Andrews an introduction to Hamilton,” 
and the encouragement came to Hamilton at an opportune 
‘moment, for he wrote :— 
“Tt was useful to me to have my attention recalled to the 
whole subject of the Quaternions, which I had been almost try- 
ing to forget, partly under the impression that nobody cared or 
would soon care about them. The result seems likely to be 
that I shall go on to write some such ‘ Manual,’ but necessarily 
a very short one.” 
The ‘‘ Manual” thus foreshadowed became the voluminous 
treatise ‘‘ The Elements of Quaternions.” 
Those interested in the future of Quaternions will have wel- 
comed the new edition of this work brought out by the present 
occupant of Hamilton’s Chair, Prof. Charles Joly, who has 
himself also added some remarkable developments to one 
branch of the subject, the Theory of the Linear Vector 
Equation. 
Hamilton’s Quaternions may be viewed in two lights, as a 
‘development of the logic and philosophy of symbols in their 
relation to space of three dimensions and also as an instrument 
of research in Geometry and Physics. In the former aspect, 
the Quaternions will ever remain a splendid monument of the 
imagination and genius of its inventor. In the latter point of 
view, that is, when we come to regard it as a working calculus, 
it would be premature as yet to fix the place it will ultimately 
occupy. 
A few years after Hamilton had entered upon his scientific 
career, James MacCullagh won his Fellowship in Trini'y College. 
After an interval of three years he was appointed Professor of 
Mathematics, and eight years later succeeded Dr. Lloyd in the 
Chair of Natural Philosophy. It would be difficult to over- 
estimate the stimulating effect of MacCullagh’s lectures as 
Professor upon the Mathematical School. Many of those whose 
names stand out afterwards—such men as Jellett, Michael and 
“William Roberts, Haughton, Townsend and _ our present 
honoured Provost—were MacCullagh’s pupils. To the present 
day the tradition still lingers in Trinity College of the impression 
MacCullagh made upon the minds of those with whom he came 
in contact. 
When, passing from his influence as a teacher, we come to 
examine his own original work, we find that this naturally 
divides itself into two departments, the first embracing Geometry 
and that part of the field of Mathematical Physics which most 
resembles Geometry, that in which the fundamental principles 
are entirely agreed upon; the second his work in Physical 
Optics, where he has to imagine new principles which, mathe- 
matically developed, should correlate the empirical laws hitherto 
obtained and be capable of verification by experiment. 
Of the first class we have his studies in ‘‘ Surfaces of the 
Second Degree.” The most striking result he here obtained 
was the discovery of the modular generation of the quadric, 
thus extending to surfaces the focus-and-directrix property of the 
conic in plano. We are also indebted to him for some very 
elegant theorems in the theory of confocal quadrics, a subject 
to which he devoted much attention. He likewise gave a course 
of lectures containing a masterly discussion and geometrical pre- 
No. 1715, VOL. 66] 
sentment of the motion of a rigid body round a fixed point not 
acted on by external forces. 
At the very outset of his career as an original author he 
seems to have been attracted by the theory of Light. To 
understand the ardour with which MacCullagh and his contem- 
poraries devoted their mathematical powers to Physical Optics, 
we must endeavour to recall the circumstances of the time. The 
celebrated memoirs of Fresnel had recently appeared. In these he 
had proved, following Young, that the ethereal vibrations which 
constitute Light must be in the plane of the wave-front ; that a 
beam of polarised light was simply a system of parallel waves in 
which these transverse vibrations were all in one direction. He 
had applied the theory of the ellipsoid to prove that there were 
three directions in a crystal in which the restitution-force coin- 
cided with the direction of the vibrations ; that in the plane of 
every wave there are two directions along which, if a particle 
vibrate, the component of the restitution-force resolved in the 
plane of the wave will be along the direction of displacement. 
He had also from these principles deduced the equation of his 
famous wave-surface. 
Tow much the work of Fresnel filled the imagination of 
scientific men in those days may be seen from the enthusiastic 
language which the sober-minded Dr. Humphrey Lloyd allows 
himself to use about him in his valuable report on Physical 
Optics, which he wrote for this Association in 1834. 
In passing I would say that the name of Fresnel reminds us 
of the loss Science, and especially this Section, has sustained 
since we last met in the death of that illustrious French physicist 
who devoted his life with such ardour and success to the same 
field of research—Alfred Cornu. Those of us who had the 
privilege of being present will recall with a sad pleasure the 
beautiful address he gave us in Cambridge on the Wave Theory 
of Light on the occasion of Sir George Stokes’ jubilee. 
Fresnel in his analysis had assumed that when the molecules of 
the ether are disturbed by the passage of a wave the force of 
restitution acting upon a molecule depends upon that molecule’s 
absolute displacement. Cauchy and Neumann and, in England, 
Green, improved on Fresnel’s reasoning, making this force de- 
pend, not on the absolute, but on the relative displacement ; all 
these physicists, however, worked on the lines of endeavouring 
to form an explanation of the propagation of the waves of Light, 
by treating them as the waves in an elastic medium, akin in its 
properties to a solid medium in which the stresses depend on 
the deformation of the elements. 
MacCullagh agreed with these others in making the forces of 
restitution depend on the relative displacements as expressed 
through a certain function V, which represented the potential 
energy of the medium. In the further development of the 
theory he, however, diverges from them and adopts a line of 
his own. Struck by the significance of the fact, to which he 
seems to have been the first to direct attention, that the vector 
whose components are 
I (2 dw) (32 du\ 1(du dv 
2\dz dy ) 2\dx dz ) Gaoes) 
which we now, of course, know as the vector of molecular 
rotational displacement, was, so to speak, a physical vector, in- 
dependent of the choice of our axes of coordinates, he was led 
to the idea of choosing for the form of V that of a homogeneous 
quadric in these three components. It must be admitted that 
the reasoning by which he attempts to prove the necessity of 
this assumption is eminently unsatisfactory, and that the 
assumption itself lay open to an apparently fatal objection urged 
later by Stokes, that of neglecting to secure the equilibrium of 
the element of the medium quoad moments. 
Having, however, adopted this form of V. MacCullagh proceeds 
(making the assumption that while the elasticity of the medium 
varied the density was everywhere the same), by processes of remark- 
able elegance and simplicity, to develop the laws of wave pro- 
pagation in a crystal, thus verifying the wave-surface of Fresnel, 
while at the same time he found himself able to satisfy com- 
pletely the requirements at the limits. He could also point to 
experience, ¢.g., the experiments of Brewster and Seebeck, as 
justifying the simple and beautiful laws which he had succeeded 
in obtaining. 
Nevertheless the force of Stokes’ objection was felt to be so 
strong that one who reviewed the subject, say thirty years ago, 
would have regarded MacCullagh’s work in Optics as presenting 
indeed opportunities for beautiful mathematical developments, 
but lacking sound physical basis. 
