544 REPORT— 1897. 



engineer, and the demands that arise in the practice of the profession, are such as to 

 force gradually a complete change of view. When I look into the text-books that 

 he uses, it seems to me a necessity that an engineer should now possess a mathe- 

 matical skill and knowledge in some directions which, not so very long since, 

 could not freely be found among the professional mathematicians themselvps. 

 And as this change is gradually effected, perhaps the practical man will gradually 

 change his estimate of the scope of mathematical science. 



I pass from the practical men to some of the natural philosophers. Many of 

 them though certainly far from all of them, expound what they consider proper 

 and economical limits to the development of pure mathematics. Their wisdom 

 gives varied reasons ; it speaks in tones of varied appreciation ; but there can be 

 no doubt as to its significance and its meaning. Their aim is to make pure mathe- 

 matics not indeed the drudge, but the handmaid of the sciences. The demand 

 requires examination, and deserves respectful consideration. There is no question 

 of (Tivinc or withholding help in furthering, in every possible fashion and with every 

 possible facility, the progress of natural philosophy ; there is no room for difference 

 upon that matter. The difference arises when the opinion is expressed or the 

 advice is tendered that the activity of mathematicians and all their investigations 

 should be consciously limited, and directed solely and supremely, to the assistance 

 and the furtherance of natural philosophy. 



One group of physicists, adopting a distinctly aggressive attitude in imposing 

 limits so as to secure prudence in the pursuit of pure mathematics, regard the 

 subject as useful solely for arriving at results connected with one or other of the 

 branches of natural philosophy ; they. entertain an honest dislike, not merely to 

 investigations that do not lead to such results, but to the desirability of carrying out 

 such investigations; and some of them have used highly flavoured rhetoric in express- 

 ing their dislike. It would be easy — but unconvincing— to suggest that, with dut 

 modifications in statement, they might find themselves faced with the necessity of 

 defending some of their own researches against attacks as honestly delivered by 

 men absorbed in purely practical work. But such a suggestion is no reply, for it 

 does not in the least touch the question at issue ; and I prefer to meet their con- 

 tention with a direct negative. 



By way of illustration let me take a special instance : it is not selected as being 

 easier to confute than any other, but because it was put in the forefront by one of 

 the vigorous advocates of the contention under discussion — a man of the highest 

 scientific distinction in his day. He wrote : ' Measured [by the utility of the 

 power they give] partial differential equations are very useful, and therefore stand 

 very high [in the range of pure mathematics] as far as the second order. They 

 apply to that point in the most important way to the great problems of nature, 

 and are worthy of the most careful study. Beyond that order they apply to 

 nothing.' This last statement, it may be remarked, is inaccurate ; for partial 

 differential equations, of an order higher than the second, occur — to give merely a 

 few examples — in investigations as to the action of magnetism on polarised light, in 

 researches on the vibrations of thick plates or of curved bars, in the discussion of 

 such hydrodynamical questions as the motion of a cylinder in fluid or the damping 

 of air-waves owing to viscosity. 



Putting this aside, what is more important is the consideration of the 

 partial ditierential equations of the second order that are found actually to occur 

 in the investigations. Each case as it arises is discussed solely in connection with 

 its particular problem ; one or two methods are given, more or less in the form 

 of rules ; if these methods fail, the attempt at solution subsides. The result 

 is a collection of isolated processes, about as unsatisfactory a collection as is the 

 chapter labelled Theory of Numbers in many text-books on algebra, when it is 

 supposed to represent Ihat great branch of knowledge. Moreover, this method 

 suflers from the additional disadvantage of suggesting little or no information about 

 equations of higher orders. 



But when the equations are considered, not each by itself but as ranged under 

 a whole system, then the investigation of the full theory places these processes in 

 their proper position, gives them a meaning which superficially they do not 



