508 KEPORT— 1902. 



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 MacCullagh in the study of quadric sur- 

 faces, and it fell to his lot to discover some most remarkable theorems on the 

 relations of the geodetics on the surface to the Hues of curvature; theorems in- 

 deed to which the author would have been justified in applying words which Gauss 

 used of a great theorem of his own : 



' Theoremata quae ni fallimur ad elegantissima referenda esse videntur.' 



Joachimsthal had shown that the first integral of the equation of the geodetics 

 en 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 nmbilics 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 oji 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 i(PAB) tan i(PBA) = const, 

 or tan"^(PAB)/tan K^'-t^^) ^ const. 



80 precisely the same relation holds for a line of curvature on the quadric, re- 

 placing the foci by the umbilics and the right lines by geodetics. 



Sir Andi-ew Hart made a valuable contribution to the subject by investigating 

 the relation between the angles which an umbilicar 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 qj and to' be these angles, 



-^^ -" . can be expressed by means of complete elliptic integrals independent 

 tan |a) 



of a. 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 ^co 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 relations by means of their sources, and in the theory of 

 Abelian integrals his construction (following the method of Jacobi) of a Trigono- 

 metry of the hyperelliptic functions. 



His brother William Roberts is perhaps best known for some of the investi- 

 gations he carried out by means of elliptic coordinates. For example, he applied 



