TRANSACTIONS OP THE SECTIONS. 5 



fail to be regarded coldly by many whose attention bad been principally directed 

 to special problems in physics. And one at least of my hearers will doubtless, with 

 myself, recollect the unrestrained censure which, in the midst of a most hearty 

 greeting, the late astronomer of Turin would pour upon the labours of any disciple 

 of the modern school who chanced to visit him. The promoters, however, of this 

 science, sure of their footing, and confident that nothing which could lead to results 

 of such a remarkable character or of such great generality, that nothing which 

 could unite, correlate, and simplify the apprehension of such numberless disjecta 

 membra of analysis — confident that no such method would in the end prove useless 

 or unmeaning in the interpretation of nature — pursued their investigations ; and a 

 very short time has justified their firmness, by witnessing the new algebra reaching 

 out and indissolubly connecting itself each year with fresh branches of mathematics. 

 The theory of equations has almost become new through it ; algebraic geometry 

 has been transfigured in its light ; the calculus of variations, molecular physics, and 

 mechanics have all felt its influence. 



The memoirs of Cay ley on quantics, those of Sylvester on the calculus of forms, 

 have become classical. Intimately connected with this subject is the theory of 

 numbers, which, at the hands of some leading analysts, principally German mid 

 French, has recently received such large extension. One peculiarity, but that of a 

 very general character, which distinguishes some of the modern from the older 

 methods, consists in the introduction of variable quantities into the expressions — in 

 other words, in bringing the processes of continuous to bear upon the properties of 

 discrete quantity. Bat into this it is unnecessary to enter in any detail, as we 

 have already in our volumes the very able and comprehensive reports by Professor 

 Smith, of Oxford. We are now anxiously expecting his final communication, not 

 only because we shall then have before us a survey of the whole subject brought 

 down to the present time, but still more because we trust that the author may then 

 find leisure to complete the original work upon the theory of numbers upon which 

 it is understood that he has been engaged for many years, and to which the reports 

 in question form only a prelude. 



The tendency which is here exhibited of some common principle running through 

 various subjects, and bringing them into connexion, reappears in the differential 

 resolvents of Cockle, Ilarley, and others, and in the transcendental solution of 

 equations which has been effected on the Continent. In both eases a relation is 

 established between ordinary algebraic equations and the differential calculus — in 

 the one with linear differential equations, in the other with a simple integration. 

 Some future developments will, perhaps, throw further light upon the ultimate issue 

 of these processes. 



The calculus of operations, or of symbols, as it has been also called, whereby the 

 symbols of operation are separated from those of quantity, has for some years been 

 in use among analysts in this country. And although no very remarkable step has 

 recently been made, or is perhaps to be expected, in this field, still some consider- 

 able progress has been effected towards completing the algebra, or laws of combi- 

 nation, of these non-commutative symbols. 



It would occupy too much time to touch upon the many more subjects which 

 suggest themselves, but it would be impossible to pass over without mention tin; 

 important contributions to the theory of differential equations, and in particular of 

 those which occur in mechanics, by the late Professors Jacobi and Boole (in 

 whose deaths mathematical science has sustained so great losses), and, secondly, 

 the extension which the theories of elliptic and Abelian functions have received at 

 the hands of Riemann, Hermite, Weierstrasse, Clebsch, and others. The last-men- 

 tioned mathematician has brought the subject of Abelian functions to bear in a 

 most remarkable and unexpected manner upon algebraic geometry. 



I will allude to only one more instance of modern generalizations — namely, the 

 conception of imaginary quantities introduced alike into geometry and algebra, 

 one of the most fertile sources of new and important theorems. The funeral on 

 this very day of one of our most profound mathematicians — Sir "W. R. Hamilton 

 — the inventor of quaternions, invests the subject with a somewhat mournful 

 aspect on the present occasion. And here I must bring this brief and imperfect 

 sketch of recent progress in our subjects to a close. It would have been more 



