the Retarded Potentials and Huyglienss Principle. 1017 



If we now let S recede to infinity, the surface integral 

 may vanish : in that case we have the usual expression, 



*-ff(m *. (8) 





the integral extending throughout all space. 



In any case the surface integral expresses the effect at P 

 of all electricity and all field conditions outside of S and 

 constitutes the usual mathematical formulation of Huyghens's 

 Principle. 



The same proof applies at once, of course, to each com- 

 ponent of the vector potential. 



§2. Heuristic Argument for the Student. 



This proof is particularly easy to lead up to. Electrostatic 

 analogies suggest seeking a solution of (1) that will repre- 

 sent a train of spherical waves proceeding from a variable 

 point-charge, and (8) is then easily guessed as the general 

 solution (cf. Jeans, ' Electricity and Magnetism '). 



Abraham's contracting sphere is then introduced as a 

 convenient way of visualizing the solution. It is next 

 suggested, either on the basis of general reasoning or from 

 the analogy of Green's stratum, that the contracting sphere 

 ought to be able to obtain contributions of potential which 

 shall represent the effect of conditions outside S from the 

 elements of S itself. Finally, we note that the spherical 

 wave from any element of charge which produces the con- 

 tribution of this element at P at time t just keeps in contact 

 with the sphere as the latter contracts, and we conclude that 

 the contributions from all sources outside S must be capable 

 of representation as an integral expressed in terms of the 

 instantaneous field conditions over the sphere. The study of 

 the integral I then falls naturally into place. 



Even the form of I can, if desired, be heuristically obtained 

 by proceeding from the analogy of the relationship between 

 the mean electrostatic potential over a sphere and its value 

 -at the centre. 



Physical Laboratory, 

 Cornell University. 



