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XXXVIIT. Some Fundamental Concepts of Electrical Theory. 

 By H. BatemaNj 21. A., Ph.D., Lecturer in Applied Mathe- 

 matics, Johns Hopkins University, Baltimore *. 



§ 1. TISES of Force.— The idea of moving lines of 

 electric force or Faraday tubes has been used 

 in brilliant fashion by Sir Joseph Thomson f to describe 

 the processes which take place in an electromagnetic field. 

 Scientists are still undecided whether to regard the lines of 

 force as physical realities or merely as useful mathematical 

 tools. The former view of the matter implies, of course, 

 that the aether has a definite structure ; and this view 

 is adopted by Sir Joseph Thomson in some of his recent 

 papers %. 



An electromagnetic theory which is based on the idea 

 of moving lines of electric force may be developed in 

 various ways. We shall begin with a very simple theory 

 which may be regarded as a development of Sir Joseph 

 Thomson's original idea in a particular direction. 



Let the equations of the moving lines of electric force be 



X(#, y, z, t) = const., Y(#, y, z, t) = const., 



where X and Y are two uniform functions of x, y, z, and t 

 which remain constant during the motion of a line of force. 

 Let X and Y be regarded as the coordinates of a point in a 

 plane II; then to each point in the plane II there corresponds 

 a moving line of electric force and vice versa. 



Now consider a closed curve C in the x,y,z space, the 

 points of this curve being considered either at one time t or 

 at different times specified by some law t=f(x, y, z), where 

 / is a uniform continuous function. For each point P 



* Communicated by the Author. 



t Phil. Mag. [5] vol. xxxi. p. 149 (1891); 'Recent Researches in 

 Electricity and Magnetism,' Oxford (1893). See also J . H. Poynting, 

 Phil. Trans. A, vol. clxxvi. p. 277 (1885) ; W. Wien, Ann. Phys. Chem. 

 Bd. xlvii. p 327 (1892) ; H. A. Lorentz, Encyklopadie der Mathematischen 

 Wissenschaften, Bd. v. § 13 (1903) p. 119 ; Sir Joseph Larmor, Proc. 

 Int. CongT. of Math., Cambridge (1912), vol. i.; E. Cunningham, 'The 

 Principle of Relativity' (1914), Chap. xv. ; 1). K. Mallik, Phil. Mag. 

 July 1913, p. 144 ; H. Bateman, Phil. Mag. Oct. 1913 & Jan. 1914; 

 'Messenger of Mathematics,' Mav 1915 ; Amer. Journ. of Math. April 

 1915 ; W. Gordon Brown, Phil. Mag. Aug. 1915, p. 282. 



% 'Electricity and Matter,' London (1904) ; Proc. Camb. Phil. Soc. 

 vol. xiv. p. 421 (1908) ; Presidential Address, British Association, 

 Winnipeg (1909) ; Phil. Mag. Feb. 1910 & Oct, 1913. See also N. R. 

 Campbell, ' The New Quarterly ' (1909) ; ' Modern Elertrical Theory/ 

 Cambridge, 2nd edit. : H. Rateman, ' Electrical and Optical Wave- 

 Motion,' Cambridge (1915) ; Bull. Amer. Math. Soc. Feb. 1915. 



