548 REPORT— 1897. 



interesting- enough in itself, but absolutely isolated. This then suggested the 

 inverse question : What is the general law of existence of such functions if they 

 exist as more than mere casual and isolated occurrences ? and how can they all be 

 determined ? The answer to these questions led to the construction of the alge- 

 braical theory of invariants for linear transformations, and subsequently to the 

 establishment of co-variantive forms in all their classes. Next came the question 

 of determining what is practically the range of their existence : that is, is there a 

 complete finite system of such functions in each particular case ? and if there is, 

 how is it composed, when in a form that ought to admit of no farther reduction ? 

 These questions, indeed, are not yet fully answered. 



While all this development of the theory of invariants was made upon 

 these lines, without thought of application to other subjects, it was soon clear that 

 it would modify them greatly. It has invaded the domain of geometry, and has 

 almost re-created the analytical theory ; but it has done more than this, for the 

 investigations of Cayley have required a full reconsideration of the very foundations 

 of geometry. It has exercised a profound influence upon the theory of algebraical 

 equations; it has made its way into the theory of diti'erential equations; and the 

 generalisation of its ideas is opening nut new regions of the most advanced and 

 profound functional analysis. And so far from its course being completed, its questions 

 fully answered, or its interest extinct, there is no reason to suppose that a term 

 can be assigned to its growth and its influence. 



As one reference has already been made to the theory of functions of a com- 

 plex variable, in regard to some of the ways in which it is providing new methods 

 in applied mathematics, I shall deal with it quite briefly now. The theory was, in 

 efi'ect, founded by Cauchy ; but, outside his own investigations, it at first made slow 

 and hesitating progress. At the present day, its fundamental ideas may be said 

 almost to govern most departments of the analysis of continuous quantity. On many 

 of them, it has shed a completely new light ; it has educed relations between 

 them before unknown. It may be doubted whether any subject is at the present 

 day so richly endowed with variety of method and fertility of resource ; its activity 

 is prodigious, and no less remarkable than its activity is its freshness. All this 

 development and increase of knowledge are due to the fact that we face at once 

 the difficulty which even the schoolboy meets in dealing with quadratic equations, 

 when he obtains ' impossible' roots ; instead of taking the wily x as our subject of 

 operation, we take the still wilier x + y ^J — \ for that purpose, and the result is a 

 transfiguration of analysis. 



In passing, let me mention one other contribution which this theory has made to 

 knowledge lying somewhat outside our track. During the rigorous revision to 

 which the foundations of the theory have been subjected in its re-establishment by 

 Weierstrass, new ideas as regards number and continuity have been introduced. 

 With biro and with others influenced by him, there has thence sprung a new 

 theory of higher arithmetic ; and with its growth, much has concurrently been 

 effected in the elucidation of the general notions of number and quantity. I have 

 already pointed out that the foundations of geometry have had to be re-considered 

 on account of results finding their origin in the theory of invariants and co- 

 variants. It thus appears to be the fact that, as with Plato, or Descartes, or 

 Leibnitz, or Kant, the activity of pure mathematics is again lending some assist- 

 ance to the better comprehension of those notions of time, space, number, 

 quantity, which underlie a philosophical conception of the universe. 



The theory of groups furnishes another illustration in the same direction. It 

 was begun as a theory to develop the general laws that govern operations of substi- 

 tution and transformation of elements in expressions that involve a number of 

 quantities : it soon revolutionised the theory of equations. Wider ideas succes- 

 sively introduced have led to successive extensions of the original foundation, and 

 now it deals with groups of operations of all kinds, finite and infinite, discrete and 

 continuous, with far-reaching and fruitful applications over practically the whole 

 of our domain. 



So one subject after another might be considered, all leading to the same 

 conclusion. I might cite the theory of numbers, which has attracted so 



