128 



SCIENCE. 



[N. S. Vol. XVI. No. 395. 



Riemann space with a single- winding 

 line. 



The next advance was to solve a problem 

 in multiform potentials in a Riemann space 

 with two winding lines. Such a case pre- 

 sents itself in finding Green's function for 

 an infinite plane with an infinitely long 

 strip cut out. Sommerfeld has treated this 

 problem by the use of the bipolar coordi- 

 nate system 



P = log — , 



This is the system used so skillfully by 

 Maxwell in which the curves p = Oj, f = 

 C2 form two orthogonal families of circles 

 (or cylinders). The Riemann space will 

 have the straight lines corresponding tO;0= 

 ± 00 for winding lines, and the plane y = 

 for branch membrane. 



The work of obtaining solutions of our 

 differential equations on other Riemann 

 surfaces or spaces has yet to be done. The 

 difficulty lies in finding an appropriate 

 system of coordinates. This is an attract- 

 ive field and seems worthy of the atten- 

 tion of the best pure mathematicians. 



It is interesting to note that the idea of 

 obtaining a new solution by integrating 

 an old solution in the complex plane with 

 regard to a parameter seems to have 

 occurred independently to a Scotch mathe- 

 matician (J. Dougal, Proceedings Edin- 

 burgh Matli. 80c. , 1901). For instance he 

 regards the Bessel function JJkr) as a 

 function of its order n, and integrates with 

 regard to n. The Legendrian and other 

 functions may be treated in the same way. 

 New functions are thus obtained that sat- 

 isfy various boundary conditions. 



All that I have said illustrates the need 

 there is for new forms of functional rela- 

 tionship. The more new functions we can 

 invent the better; that is to say, functions 

 with new and varied characteristic pro- 

 perties. "We look to general function 



theory to supply them. One never knows 

 how soon they may find suitable use in some 

 field of pure or physical mathematics. I 

 said at the beginning that a number of 

 physical problems are at a standstill for 

 want of an appropriate mode of mathe- 

 matical expression. In proof of this I may 

 here quote the words of a few experts in 

 different lines of work. 



Lord Rayleigh says,* "When the fixed 

 boundary of a membrane is neither straight 

 nor circular, the problem of determining 

 its vibrations presents difficulties which in 

 general could not be overcome without the 

 introduction of functions not hitherto dis- 

 cussed or tabulated. A partial exception 

 must be made in favor of an elliptic 

 boundary." 



I may note here that Mathieu solved the 

 problem of the elliptic membrane by trans- 

 forming the differential equation to elliptic 

 coordinates ( f , rj), so that one coordi- 

 nate S would be constant on an elliptic 

 boundary, and then satisfying the equation 

 by means of a product function 



u= p(?) •vH'''/). 

 making ^ vanish on the boundary. This 

 method might seem promising for other 

 boundaries ; but Michell has proved that the 

 elliptic transformation is the only one that 

 leads to an equation capable of being satis- 

 fied in the product form.t 



Lord Rayleigh says in another place : X 

 "The problem of a vibrating rectangular 

 plate is one of great difficulty, and has for 

 the most part resisted attack. * * * The 

 case where two opposite edges are free while 

 the other two are supported has been dis- 

 cussed by Voigt. ' '§ 



In connection with air vibration he says ; 

 ' ' The investigation of the conductivity for 

 various kinds of channels is an important 



*• Theory of Sound/ Vol. 1, p. 343 (2d ed.). 



t M^'ssenycr of Mathematics, 1890. 



J 'Theory of Sound,' Vol. 1, p. 372. 



§ Gottingen Nachrichteii, 1893. 



