37S 



NA rURE 



[August 19, 1897 



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 

 algebraical theory of invariants for linear transformations, and 

 subsequently to the establishment of covariantive 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 further 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 recon- 

 sideration 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 differential equations ; 

 and the generalisation of its ideas is opening out 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 complex 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 

 effect, founded by Cauchy ; but, outside his own investigations, 

 it at first made slow and hestitating 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 difiiculty 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 him 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 covariants. 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 assistance 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 substitution and trans- 

 formation of elements in expressions that involve a number of 

 quantities : it soon revolutionised the theory of equations. 

 Wider ideas successively 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 many of the keenest intellects among 

 men, and has grown to be one of the most beautiful and 

 wonderful theories among the many in the wide range of pure 

 mathematics ; or without entering upon the question whether 

 geometry is a pure or an applied science, I might review its 

 growth alike in its projective, its descriptive, its analytical, and 

 its numerative divisions ; or I might trace the influence of the 



NO. 



[451, VOL. 56] 



idea of continuity in binding together subjects so diverse as 

 arithmetic, geometry, and functionality. What has been said 

 already may, however, suffice to give some slight indication of 

 the vast and ever- widening extent of pure mathematics. No 

 less than in any other science knowledge gathers force as it 

 grows, and each new step once attained becomes the starting- 

 point for steady advance in further exploration. Mathematics 

 is one of the oldest of the sciences ; it is also one of the most 

 active, for its strength is the vigour of perpetual youth. 



In conclusion, a fe\^ words are due to the personal losses 

 caused since our last meeting. It is but little more than two 

 years since Cayley passed away ; his life had been full of work, 

 unhasting and unresting in the almost placid course of his great 

 mental strength. While Cayley was yet alive, one name used 

 to be coupled with his when reference was made to English pure 

 mathen)atics ; the two great men were regarded as England's 

 not unworthy contribution to the exploration of the most abstract 

 of the sciences. These fellow-workers, diverse in temperament, 

 in. genius, in method, were bound by a friendship that was ended 

 only by death. And now Sylvester too has gone ; full of years 

 and honours ; though he lived long, he lived young, and he was 

 happily active until practically the very end. Overflowing with 

 an exuberant vitality alike in thought and work, he preserved 

 through life the somewhat rare faculty of instilling his 

 enthusiasm into others. Among his many great qualities, not 

 the least forcible were his vivid imagination, his eager spirit, 

 and his abundant eloquence. When he spoke and wrote of his 

 investigations, or of the subject to which the greater part of his 

 thinking life had been devoted, he did it with the fascination ot 

 conviction ; and at times— for instance, in his presidential 

 address to this Section at Exeter in 1869 — he became so 

 possessed with his sense of the high mission of mathematics, 

 that his utterances had the lofty note of the prophet and the seer. 

 One other name must be singled out as claiming the passing 

 tribute of our homage ; for, in February last, the illustrious and 

 venerable Weierstrass died. He was unconnected with our 

 Association ; but science is wider than our body, and we can 

 recognise and salute a master of marvellous influence and 

 unchallenged eminence. 



Thus, even to mention no others, pure mathematics has in a 

 brief period lost three of the very greatest of its pioneers and 

 constructors who have ever lived. We know their genius ; and 

 the world of thought, though poorer by their loss, is richer by 

 their work. 



Tho' much is taken, much abides, and tho' 



We are not now that strength which in old days 



Moved earth and heaven ; that which we are, we are : 



One equal temper of heroic hearts, 



Made weak by time and fate, but strong in will 



To strive, to seek, to find, and not to yield. 



Knowledge cannot halt though her heroes fall : the example of 

 their life-long devotion to her progress, and the memory of 

 their achievements, can inspire us and, if need be, can stimulate 

 us in realising the purpose for which we are banded together 

 as an Association — the advancement of science. 



SECTION B. 



chemistry. 



Opening Address by Prof. William Ramsay, Ph.D., 



LL.D., Sc.D., F.R.S., President of the Section. 



An Undiscovered Gas. 

 A SECTIONAL address to members of the British Association 

 falls under one of three heads. It may be historical, or actual, 

 or prophetic ; it may refer to the past, the present, or the future. 

 In many cases, indeed in all, this classification overlaps. Your 

 former Presidents have given sometimes a historical introduction, 

 followed by an account of the actual state of some branch of 

 our science, and, though rarely, concluding with prophetic 

 remarks. To those who have an affection for the past, the his- 

 torical side appeals forcibly ; to the practical man, and to the 

 investigator engaged in research, the actual, perhaps, presents 

 more charm ; while to the general public, to whom iiovelty is 

 often more of an attraction than truth, the prophetic aspect 

 excites most interest. In this address I must endeavour to tickle 

 all palates ; and perhaps I may be excused if I take this oppor- 

 tunity of indulging in the dangerous luxury of prophecy, a luxury 

 which the managers of scientific journals do not often permit 

 their readers to taste. 



