TRANSACTIONS OF SECTION A. 545 



exhibit, and indicates the way in which each solution satisfies the general con- 

 ditions of existence of a solution. For the full theory of partial differential 

 equations of the second order in, say, two independent variables establishes the con- 

 ditions of existence of a solution, the limitations upon the conditions which make 

 that solution unique, the range of variation within which that solution exists, the 

 modes of obtaining expressions for it when it can be expressed in a finite form, and 

 an expression for the solution when it cannot be expressed in a finite form. Of 

 course, the actual derivation of the solution of particular equations is dependent 

 upon analytical skill, as is always the case in any piece of calculating work ; but 

 the general theory indicates the possibilities and the limitations which determine 

 the kind of soliition to be expected. But not only does the general theory effect 

 much by way of co-ordinafing isolated processes — and, in doing so, lead to new 

 results — but it gives important indications for dealing with equations of higher orders, 

 and it establishes certain theorems about them merely by simple generalisations. 



In fact, the special case quoted is one more instance, added to the many 

 instances that have occurred in the past, in which the utilitarian bias in the 

 progress of knowledge is neither the best stimulus nor in the long run the most 

 effective guide towards securing results. It may be — it frequently is — at first 

 the only guide possible, and for a time it continues the best guide, but it does not 

 remain so for ever. It would be superfluous, after Cayley's address in 1883, to 

 show how branches of mathematical physics, thus begun and developed, have 

 added to knowledge in their own direction; they have suggested, they have even 

 created, most fascinating branches of pure mathematics, which, when developed, 

 have sometimes proved of reciprocal advantage t9 the source from which they 

 sprang. But for proper and useful development they must be free from the 

 restrictions which the sterner group of natural philosophers would lay upon 

 them. 



Now I come to another group of natural philosophers who will unreservedly 

 grant my contention thus far ; who will yield a ready interest to our aims and 

 our ideas, but who consider that the possibility of applying our results in the 

 domain of physical science should regulate, or at least guide, advance in our work. 

 Some of these entertain this view because they think that possibility of early appli- 

 cation is, in the last resource, the real test of useful development ; some, because 

 they fear that the profusion of papers annually published and the bewildering 

 specialisation in each branch, are without purpose, and may ultimately lead to 

 isolation or separation of whole sections of mathematics from the general progress 

 of science. 



The danger arising from excess of activity seems to me unreal ; at any rate 

 there are not signs of it at home at the present day, and I would gladly see 

 more workers at pure mathematics, though not of course at the expense of attention 

 paid to any other branch. But for results that are trivial, for investigations that 

 have no place inorganic growth and development, or in illustration and elucidation, 

 surely the natural end is that they soon subside into mere tricks of ' curious pleasure 

 or ingenious pain.' However numerous they may be, they do not possess intrinsic 

 influence sufficient to cause evil consequences, and any attempt at repression will, 

 if successful, inevitably and unwisely repress much more. 



More attention must be paid to the suggestion that mathematicians should be 

 guided in their investigations by the possibility of practical issues. That they are 

 so guided to a great extent is manifest from many of the papers written in that 

 spirit ; that they cannot accept practical issues as the sole guide would seem 

 sufficiently justified by the consideration that practical issues widen from year 

 to year and cannot be foreseen in the absence of a divining spirit. Moreover, if such 

 a principle were adopted, many an investigation undertaken at the time for its in- 

 trinsic interest would be cast on one side unconsidered, because it does not satisfy 

 an external test that really has nothing to do with the case, and may change its 

 form of application from time to time. 



To emphasise this opinion that mathematicians would be unwise to accept 

 practical issues as the sole guide or the chief guide in the current of their investiga- 

 tions, it may be suficient to recall a few instances from history in which the 



1897. N N 



