December 26, 1913] 



SCIENCE 



905 



of the solution giving us their periods, and 

 the nature of the function U showing the 

 phenomenon of resonance, that is to say, the 

 vibration becoming infinite when the im- 

 pressed force has the period of one of the 

 natural vibrations. The method of Poin- 

 eare leads directly to Schmidt's solution of 

 the integral equation. 



In a third paper published in the Acta 

 Mathematica in 1897 Poincare deals with 

 what he calls Neumann's problem, which 

 he defines as follows: To find a potential 

 of a double layer whose limiting values in- 

 side and outside the surface are denoted 

 by V and V, and which satisfies the equa- 

 tion at the surface, 



v—r'=:\(v + v) -1-2*, 



where A is a parameter. If A^ — 1 this 

 reduces to the interior Diri chiefs problem 

 "F = <1) and if A^l to the exterior problem 

 y = — $. By means of a development in 

 powers of the parameter X a solution is ob- 

 tained by successive approximations which 

 is proved to converge. One of the most im- 

 portant results of this paper is the demon- 

 stration of the existence of a series of what 

 he calls fundamental functions which have 

 the property of being potentials of simple 

 layers, and 



in terms of which he deems it probable 

 that any function on the surface may be 

 developed, so that when these functions are 

 known Dirichlet's problem may be solved. 

 These reduce for the sphere to spherical 

 harmonics and for the ellipsoid to Lame's 

 functions, and they are the characteristic 

 functions belonging to integral equations. 

 Let us now turn to a different field. In 

 1893 attention was called by Poincare to 

 an equation which has become famous, 

 called by him the equation of telegraphists. 

 This equation 



had been introduced before by Kirchhoff 

 and Heaviside, but its physical interpreta- 

 tion had not been emphasized. If the first 

 term is lacking it reduces to Fourier's 

 equation and it had been shown by Sir 

 William Thomson in 1855 that signals 

 were propagated through a submarine cable 

 in accordance with it. If the second and 

 third terms are absent the equation re- 

 duces to the equation of sound in one di- 

 mension and shows the propagation of 

 waves unchanged in form with a constant 

 velocity. The equation of telegraphists 

 may then be expected to combine the prop- 

 erties of transmission in waves and heat 

 transmission with an infinite velocity. The 

 first term arises from the consideration of 

 the self-induction of the line neglected by 

 Thomson and the second term from the re- 

 sistance which can generally not be ne- 

 glected. By the simple method of the as- 

 sumption that IV can be represented as a 

 Fourier's integral, after taking out an ex- 

 ponential factor, so that 



Poincare obtains the solution 



U= r en' le cos WF^ + e,?HL4iI^l dq, 



•^-«' L V92 - 1 J 



which he shows by an application of the 

 theory of functions to depend upon a Bes- 

 sel function. The remarkable physical re- 

 sult is that while the disturbance, like the 

 sound wave, is propagated with a finite 

 velocity, after it has passed over a given 

 point it leaves a residue or trail which 

 gradually dies away like heat. In a later 

 paper he discussed the effect on the teleg- 

 raphist's equation of terminal conditions of 

 a complicated sort necessitated by the em- 

 ployment of receiving apparatus. 



Probably the favorite subject in mathe- 



