PRESIDENTS ADDRESS — SECTION A. 17 



habitation ; but it was his training in the School of Mines that 

 enabled Poincare to make the important contributions to Mathematical 

 Physics which will always be associated with his name. Sommerfeld, 

 whose name we meet so often in the discussions upon Wireless Tele- 

 graphy, has this double qualification. A profound mathematician, 

 he has also the " physical instinct " of which Berry spoke. As to 

 Hilbert, I cannot be certain ; but my impression is that he also has 

 had the advantage of a training in Physics. The wide range of his 

 work must be a perpetual source of wonder to other mathematicians. 

 Like Poincare, we find him deep in the discussion of the foundations 

 of Geometry, and shedding fresh light upon the real meaning of the 

 Non-Euclidean Geometries. He has also contributed to various 

 departments of Higher Algebra and Analysis. And he was the first 

 to grasp the true significance of Fredholm's discovery of the solution 

 of the Integral Equations which now bear his name. Hilbert 

 established the connexion between the Theory of Integral Equations 

 and that of Quadratic forms. Then he applied his discovery to the 

 differential equations of Mathematical Physics, and united in one 

 general theorem all the different questions of the expansion of an 

 arbitrary function in series, whether the terms be trigonometrical 

 functions, Bessel's Functions, Spherical Harmonics, Ellipsoidal 

 Harmonics, or Sturm's Functions. And the last two chapters of the 

 book, in which he has brought togetner his contributions to this new 

 branch of analysis, are devoted — one to its applications to the Theory 

 of Functions, and the other to its applications to the Calculus of 

 Variations, Geometry, Hydrodynamics, and the Kinetic Theory of 

 Gases. He now seems to have turned to Mathematical Physics, and 

 he has recently lectured on the Kinetic Theory of Gases. One of his 

 courses during last summer Semester was upon the Mathematical 

 Foundations of Physics. And during this .winter Semester he has 

 lectured upon the Theory of Partial Difierential Equations, and 

 continued his former course on the Foundations of Physics, his 

 Seminar being given up to discussions on similar topics. 



Another German mathematician, Kneser, in his book " On the 

 Application of Integral Equations to Mathematical Physics," and in 

 his published writings upon the same subject, seems to have stepped 

 right into the region which we might have expected the Cambridge 

 School to have already occupied, if it had not broken with the traditions 

 of its past. And the same may be said of part of the work of Stekloff, 

 the Russian mathematician. 



There are, of course, many modern developments of Pure Mathe- 

 in.atics with a close bearing upon the problems which meet us in 

 Applied. In some of my own investigations I have had occasion to 

 use the Theory of Functions of a Complex Variable, and Riemann's 

 Surfaces and Space. A year or so ago some work on Non-Euclidean 

 Geometry compelled me to put aside these questions, and it is only 



