568 Mr. C. E. Weatherburn: Problems in 



themselves to the reader. The Tables given, with the 

 formula replacing them when a > 80, are sufficient to cover 

 the whole range of necessary calculations. A spectrum 

 which contains even so many lines as 150 can be explored 

 for series in this manner in a very short time, especially when 

 regard is paid to the fact that the intensities of the lines, as 

 measured experimentally on any system, must diminish as we 

 proceed towards the limit of the series. 



LIX. Problems in Electrostatics and the Steady Flow of Elec- 

 tricity under the exponential potential e~ kr /r. By C. E. 

 Weatherburn, M.A. {Cantab.), M.A., B.Sc. (Sydney); 

 Lecturer in Mathematics and Physics, Ormond College, 

 University of Melbourne* . 



Introduction. 



ONE of the difficulties with which the present day physicist 

 is confronted, is that of assigning a definite law of 

 action at infinitesimal distances between two elements of 

 electricity. Though Newton's law is universally admitted 

 for finite distances, there is considerable doubt as to its 

 validity for small ones. In other branches of physics, too, 

 there is evidence that the action between particles very close 

 together does not conform to the inverse square law. 



We are therefore justified in investigating laws of attrac- 

 tion with potentials different from that of the inverse 

 distance. In particular, the exponential potential e~ hr lr, in 

 which k is a positive constant, has peculiar claims upon our 

 attention, for it was shown by Neumann f at the close of the 

 last century that only under a potential of the form 



2A*^ 



k r 



is the electric equilibrium possible within a conductor. 

 The exponential potential includes the Newtonian as a 

 particular case, and Green's potential J 



Pi 



—2 



i=l 



* Communicated by the Author. 



tAUgemeine Untersuchungen uber das Newton'sche Priucip der 

 Fernwirkungen, Cap. 2. Teubner, Leipzig, 1896. 



J Green, " Math, investigations concerning the laws of the equilibrium 

 of fluids, &c." Trans. Camb. Phil. Soc. 1833. 



