July 25, 1902.] 



SCIENCE. 



129 



part of the theory of resonators, but in all 

 except a very few cases the accurate solu- 

 tion of the problem is beyond the power 

 of existing mathematics."* 



Professor E. L. Bro%vn in his report on 

 hydrodynamics presented to the Boston 

 meeting- says : "No problem of discontinu- 

 ous motion in three dimensions has yet been 

 solved. The difficulty is one which can be 

 easily appreciated. The theory of func- 

 tions deals with a complex of the form 

 oi-\-iy and this suits all problems in two 

 dimensions. But little has been done with 

 a vector in three dimensions. Perhaps the 

 paper on Potentials by Sommerfeld in the 

 Proceedings of the London Mathematical 

 Society last year may have some bearing 

 on the problem ; it is in any case worth seri- 

 ous study. The subject of discontinuous 

 motion was set for the Adams prize in 1895. 

 A solution for a solid of revolution was 

 asked for, and it was generally supposed 

 that the circular disc would be the easiest 

 to attempt. No solution was sent in. One 

 prominent mathematician who has aided 

 considerably in the development of hydro- 

 dynamics mentioned that he had worked 

 for six months and had obtained absolutely 

 nothing. A magnificent reception there- 

 fore awaits the first solution." 



Mr. Hayford writes (in Science, 1898) : 

 ' The most important tidal problem before 

 us is that of determining the relation be- 

 tween the boundaries (bottoms and shores) 

 and the modification produced by them on 

 the tidal wave.' 



Professor Webster, in his report on re- 

 cent progress in electricity and magnetism, 

 presented to the Boston meeting, says: 

 ' ' The problem of electrical vibrations in a 

 long spheroid is next to be attacked, and 

 then perhaps on surfaces obtained by the 

 revolution of the curves known as cyclides. 

 The introduction of suitable curvilinear 



* ' Theory of Sound,' Vol. 2, p. 175. 



coordinates into the partial differential 

 equations will lead us in the ease of the 

 spheroid to new linear differential equa- 

 tions, analogous to, but more complicated 

 than, Lame's, and will necessitate the in- 

 vestigation of new functions and develop- 

 ments in series." 



Dr. Webster also commends to the atten- 

 tion of pure mathematicians the various 

 differential equations which are to be found 

 in Heaviside's electrical papers; more es- 

 pecially the question of existence theorems. 



I may mention here that Hilbert in a re- 

 cent volume of the Archiv^' suggests the 

 question of proving an existence theorem 

 for the solution of any differential equa- 

 tion subject to assigned boundary condi- 

 tions. 



Even a partial treatment of any one of 

 these problems might open up new rela- 

 tionships, and widen the intellectual hori- 

 zon. It is a hopeful sign that several pure 

 mathematicians are turning their attention 

 to such questions. Speaking at the Chi- 

 cago Mathematical Congress in 1893, Pro- 

 fessors Klein and Webster deplored the 

 growing separation of the pure and phys- 

 ical branches of mathematics, and pointed 

 out the great loss that would result to each 

 of the divergent branches. The recent in- 

 creased attention to mathematical history 

 has enforced this opinion. The influence 

 of Klein, Poincare, Weber and others has 

 been helpful as a corrective, on the con- 

 tinent of Europe. The British Universi- 

 ties have steadfastly treated mathematical 

 physics as an organic part of mathematical 

 discipline. The same statement could not 

 be made with regard to all of the American 

 Universities ; but there are many signs of 

 improvement. With a true historical in- 

 stinct, this Section of the Association, and 

 its ally, the American Mathematical So- 

 ciety, have exerted their influence for an 



* Archiv Math, und Phys., 1901, p. 229. 



