N. Y. AO'^DEMY 

 OF SClhlMGeS 



PROCEEDINGS 



OP 



THE ROYAL IRISH ACADEMY 



PAPERS READ BEFORE THE ACADEMY 



I. 



A THEOREM OX MOVINQ DISTEIBUTIONS OF ELECTRICITY. 



By ARTHUR W. COXWAY, M.A., F.R.U.I., 



Professor of Mathematical Physics, University College, Dublin. 



Read NoTEMBER 12. Ordered for publication December 17,1906. Piitlished Jaxi-aet 31, 1907. 



The field of force due to a moving electron is defined by a scalar potential -i^, 

 and a vector potential (F, G, H), the forces being given by the following 

 equations : 



The electric force {X, F, ^) = _ fl 1 IV^ - 1 {F, G, H). 



\cx dy ozj ' (jt 



The magnetic force («, /3, 7) = curl {F, G, H). 



The potential functions are formed as follows : — Let fi(t), fi{i), fi{i) ^^e 

 the Cartesian coordinates of the moving electron, the charge of which is e, 

 and let t be the real root less than t of the equation 



V\t - rf = C. -Mr)y + {y -Mr)Y + {Z -Mr))\ 



where V denotes the speed of radiation ; then, if we denote by the 

 expression 



V\t - r) - {:c -Mr))f/(r) - (y - Mr)) Mr) - {z -/3(r))//(r), 



we have 



■^ = eVY\ (F, G, H) = (//(r), //(r), //(r)) eVcl>-\ 



We may also notice that can be written 



{ni)-^ldu[V\t - «y- - {X -M2c)f - (y -A{n)f - (. -/3(«))T, 



where u is complex, and the integration is taken over a closed path surrounding 



R. I. A. PROC, VOL. XXVII., SECT. A. [IJ 



