504 REPORT— 1902. 



' It was useful to me to have my attention recalled to the whole subject of 

 the Quaternions, which I had been almost trying to forget, partly imder 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 welcomed the new 

 edition of this work brought out by the present occupant of Hamilton's Chair, 

 Professor 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 

 regrard 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 Trinity College. After an interval of three 

 years he was appointed Professor of Mathematics, and eight years later suc- 

 ceeded Dr. Lloyd in the Chair of Natural Philosopliy. It would be difficult to 

 overestimate the stimulating efl'ect of MacCuUagh's lectures as Professor upon the 

 Mathematical School. Many of those whose names stand out afterwards — such 

 men as Jellett, Michael and "William Iloberts, Haughton, Townsend, and our 

 present honoured Pro\'Ost — were MacCuUagh'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, tbat in which tlie fundamental priuciples are entirely 

 agreed upon ; the second his work in Physical Optics, where he has to imagine 

 new principles which, mathematically 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 piano. 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 geo- 

 metrical presentment 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 

 MactJullagh and his contemporaries 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 lestitution-force coincided witli 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 reistitutiou-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. 



How much the work of Fresnel filled the imagination of scientific men in 



