482 
NATURE 
| SEPTEMBER 11, 1902 
When the Association met in’ Belfast twenty-eight years ago 
Dr. Jellett occupied this Chair, and at the close of his Address, 
in which he took for his subject certain fresh applications of 
Mathematical Analysis to Physical Science, he touched upon 
these very researches in which he was at the time engaged. 
All old Trinity men would think this enumeration incomplete 
if it did not refer to the wonderfully active animating presence 
of Samuel Haughton. He also directed his energies in the first 
instance to the subject of Elasticity, on which he wrote several 
important memoirs, endeavouring to formulate a system of laws 
by which he might be able to explain the propagation of Light. 
But apparently discouraged by the extreme difficulty of the 
problem his versatile brain turned soon to quite other branches 
of science—to Physical Geology, then to Physiology and 
Medical Science, and in fact in his later work he passes out of 
the cognisance of Section A. 
Of the pure mathematicians trained under MacCullagh two of 
the most eminent were the twin brothers Michael and William 
Roberts. Strikingly alike in their personal appearance they 
were in my student days two of the best known figures in the 
Courts of Trinity. 
In his geometrical work Michael Roberts pursued the fruitful 
lines of research started by Chasles and followed up by Mac- 
Cullagh in the study of quadric surfaces, and it fell to his lot to 
discover some most remarkable theorems on the relations of the 
geodetics on the surface to the lines of curvature ; theorems 
indeed to which the author would have been justified in applying 
words which Gauss used of a great theorem of his own: 
**Theoremata que ni fallimur ad elegantissima referenda 
esse videntur.” 
Joachimsthal had shown that the first integral of the equation of 
the geodetics on an ellipsoid could be thrown into the well- 
known form PD = constant. Michael Roberts now showed 
that the geodetics, which issue in all directions from an umbilic, 
pass through the opposite umbilic where they meet again by 
paths of equal length ; that the lines of curvature considered 
with respect to two interior umbilics possess properties closely 
analogous to those of the plane conic with respect to its foci ; 
that if such umbilics A and B be joined by geodetics to any 
point P on a given line of curvature they make equal angles 
with such line, and consequently that as P moves along the line 
of curvature, either PA+PB or PA—PB remains constant, so 
that if the ends of a string be fastened at the two umbilics and 
a style move over the surface of the ellipsoid, keeping the string 
stretched, the style will describe a line of curvature. Another 
remarkable analogue he proved was the following : that as in a 
plane conic if a point P on the curve be joined to the foci 
A and B, 
tan 3(PAB) tan 3(PBA) = const. 
or tan 4(PAB)/tan 4(PBA) = const. 
so precisely the same relation holds for a line of curvature on the 
quadric, replacing the foci by the umbilics and the right lines by 
geodetics. 
Sir Andrew Hart madea valuable contribution to the subject 
by investigating the relation between the angles which an um- 
bilicar geodetic makes with the principal plane when it leaves 
the umbilic and when it returns to it again after going the 
circuit of the surface. He proved that if w and w’ be these angles, 
tan 30’ 
tan 4w 
tegrals independent of w. This is interesting, as it shows that 
such a geodetic is not a finite closed curve, but that it crosses 
itself over and over again at the umbilics, the successive values of 
tan 4w forming a geometric series. 
To Michael Roberts is also due much important work in the 
department of pure analysis—notably, in modern Algebra his 
method of deriving Covariants, and the investigation of their re- 
lations by means of their sources, and in the theory of Abelian 
integrals his construction (following the method of Jacobi) of a 
Trigonometry of the hyperelliptic functions. 
His brother William Roberts is perhaps best known for some 
of the investigations he carried out by means of elliptic co- 
ordinates, For example, he applied them to Fresnel’s wave- 
surface, and showed that the two sheets of the surface can be 
expressed in the simple forms 
AP+y2= a+ 027—c? and A2+p2=a2+ 2-02. 
By following the same method he succeeded also in adding an 
interesting new triple system of orthogonal surfaces to those 
already known. 
NO. 1715, VOL. 66] 
can be expressed by means of complete elliptic in- 
Richard Townsend was another of the Fellows of Trinity of 
MacCullagh’s school. He was known to us in College in my 
day as the great expositor of the new geometry of Anharmonics 
and Involution. He wrote many valuable original papers, but 
it was as a lecturer he was most remarkable. I never met a 
teacher so enthusiastic or one who seemed to enjoy teaching 
more thoroughly. 
He inspired his pupils with much of his own ardour, and it is 
greatly owing to Townsend’s influence that the old name Trinity 
had for the study of Geometry was so well kept up in his day. 
He published in the latter part of his life an extensive treatise 
on Modern Geometry, which did good service in presenting the 
subject in the light of an organised system and not as a collection 
of isolated problems. 
In this connection I must not omit to mention one of our 
most original Irish geometers of recent days, Dr. John Casey. 
Where Casey learnt his Mathematics is indeed a marvel. Up 
to middle life he was engaged in the engrossing labour of a 
schoolmaster in Kilkenny under the National Board of Educa- 
tion. It was not till he was nearly forty that by the advice of 
Townsend, to whom he used to send up some of his ingenious 
geometrical solutions, he moved up to Dublin and entered 
Trinity College. Of his original papers his best known are 
those on Bicircular Quartics and Cyclides. 
In elementary Geometry we owe to hima very elegant ex- 
tension of Ptolemy’s famous theorem that for four points, 
ABCD, on a circle AC._BD = AB.CD + AD.BC. Casey 
shows that the same equation is true if we replace the four points 
by four circles touching a common circle and the lines joining 
the points by the common tangents to the circles. He acquired 
so high a repute both as a teacher and asa writer that he was 
offered and accepted the post of Professor of Mathematics in the 
Catholic University. 
It is not yet two years since George FitzGerald was taken from 
us. The many loving tributes to his memory which appeared in 
§ the scientific journals after his death reveal to us how deep and 
widespread his loss was felt to be, but it isin Ireland this loss 
is most serious. As long as he lived and worked, our country 
could claim to own one of the foremost members of that select 
band who are endeavouring to wrest from Nature her inmost 
secrets. 
You knew how sedulous an attendant he was of the meetings 
of this Section, and Trinity College never sent you a repre- 
sentative of whom she had more reason to be prowt, for he has 
done more than any of her sons for many years to maintain the 
reputation of her scientific school. This he has brought about, 
not by his writings only, able and original as these were, but 
also by the encouragement and stimulus he gave the younger 
men he gathered round him, and the self-forgetful readiness 
with which he gave all the help he could to those who in any 
measure shared his own genuine love for science. 
You will all rejoice that we are now in possession of a volume 
containing a complete collection of FitzGerald’s scientific papers. 
T am sure he himself could not have wished for a better chron- 
icler of his life and labour than his intimate friend Dr. Larmor, 
more especially as Dr. Larmor’s own far-reaching speculations 
on the great mystery of the ether qualify him in a very peculiar 
manner to appreciate the work of his fellow-physicist. The 
admirable analysis of that work in the opening pages of this 
volume renders any further account of it on my part completely 
unnecessary. 
A few months before FitzGerald’s death there passed away 
one of his most distinguished pupils, Thomas Preston. Though 
cut off so young he had already done much work, and of a 
quality which raised high expectations of his future. His 
treatises on Light and on Heat are to be noted, not merely for 
the excellent account they give of the recent additions to the 
subjects treated, but for the thoughtful and philosophic spirit in 
which the whole is presented. It was, however, his experi- 
mental researches which most excited attention, more particu- 
larly those on the action on Light of a strong electro-magnetic 
field and the fine experiments in which he extended beyond any 
observations hitherto made the analysis of the Zeeman effect. 
Of two others I have yet to speak, and these were emphati- 
cally representatives of this city and of the College in whose 
Halls we are meeting to-day—Thomas Andrews and James 
Thomson. It would be difficult to describe adequately all the 
phases of so manifold an activity as that of Dr. Andrews. As 
one long associated with him as a colleague I would bear 
testimony to one side of his life-work—the potent influence he 
