546 REPORT — 1897. 



purely mathematical discovery preceded the practical application and was not an 

 elucidation or an explanation of ohserved phenomena. The fundamental properties 

 of conic sections were known to the Greelis in the fourth and the third centuries 

 before the Christian era ; but they remained unused for a couple of thousand 

 years until Kepler and Newton found in them the solution of the universe. Need 

 I do more than mention the discovery of the planet Neptune by Adams and 

 Leverrier, in which the intricate analysis iised had not been elaborated for such 

 particular applications .^ Again, it was by the use of refined analytical and 

 geometrical reasoning upon the properties of the wave-surface that Sir W. R. 

 Hamilton inferred the existence of conical refraction which, down to the time 

 when he made his inference, had been ' unsupported by any facts observed, and 

 was even opposed to all the analogies derived from experience.' 



It may be said that these are time-honoured illustrations, and that objec- 

 tions are not entertained as regards the past, but fears are entertained as regards 

 the present and the future. Very well ; let me take one more instance, by choosing 

 a subject iu which the purely mathematical interest is deemed supreme, the theory 

 of functions of a complex variable. That at least is a theory in pure mathematics, 

 initiated in that region and developed in that region ; it is built up in scores of 

 papers, and its plan certainly has not been, and is not now, dominated or guided 

 by considerations of applicability to natural phenomena. Yet what has turned 

 out to be its relation to practical issues ? The investigations of Lagrange and others 

 upon the construction of maps appear as a portion of the general property of 

 conformal representation ; which is merely the general geometrical method of 

 regarding functional relations in that theory. Again, the interesting and important 

 investigations upon discontinuous two-dimensional fluid motion in hydrodynamics, 

 made in the last twenty years, can all be, and now are all, I believe, deduced from 

 similar considerations by interpreting functional relations between complex vari- 

 ables. In the dynamics of a rotating heavy body, the only substantial extension 

 of our knowledge made since the time of Lagrange has accrued from associating 

 the general properties of functions with the discussion of the equations of motion. 

 Further, under the title of conjugate functions, the theory has been applied to 

 various questions in electrostatics, particularly in connection with condensers and 

 electrometers. And, lastly, in the domain of physical astronomy, some of the most 

 conspicuous advances made in the last few years have been achieved by introducing 

 into the di.scussion the ideas, the principles, the methods, and the results of the 

 theory of functions. It is unnecessary to speak in detail of this last matter, for I 

 can refer you to Dr. G. W. Hill's interesting ' Presidential Address to the American 

 Mathematical Society' in 1895; but without doubt the refined and extremely 

 difficult work of Poincare and others in physical astronomy has been possible only 

 by the use of the most elaborate developments of some pure mathematical subjects, 

 developments which were made without a thought of such applications. 



Now it is true that much of the theory of functions is as yet devoid of explicit 

 application to definite physical subjects ; it may be that these latest applications 

 exhaust the possibilities in that direction for any immediate future ; and it is also 

 true that whole regions of other theories remain similarly unapplied. Opinion 

 and divination as to the future would be as vain as they are unnecessary ; but my 

 contention does not need to be supported by speculative hopes or uninformed 

 prophecy. 



If in the range of human endeavour after sound knowledge there is one subject 

 that needs to be practical, it surely is Medicine. Yet in the field of Medicine it 

 has been found that branches such as biology and pathology must be studied for 

 themselves and be developed by themselves with the single aim of increasing 

 knowledge ; and it is then that they can be best applied to the conduct of living 

 processes. So also in the pursuit of mathematics, the path of practical utility is 

 too narrow and irregular, not always leading far. The witness of history shows 

 that, in the field of natural philosophy, mathematics will furnish more eflfective 

 assistance if, in its systematic development, its course can freely pass beyond the 

 ever-shifting domain of use and application. 



What I have said thus far has dealt with considerations arising from the 



