1891.] On the Theory of Electrodynamics. 531 



For the plane wave 



V oc exp i (mxnt), 

 it leads to the condition 



K, 



and the velocity of propagation is 



where X is the wave-length! 



The special case of 0(r) equal to zero is worth notice, as that 

 would represent a theory in..whicb the element of Neumann's in- 

 tegral, viz., ids i'd<i' cose/r, is the mutual energy of two current ele- 

 ments. When the. currents are not circuital, this leads to a con- 

 densational wave of ; the -type here given. .. 



(ii) If 0(r) = Cr (von Helmholtz's hypothesis) the equation 

 becomes 



denoting undulatory propagation with constant velocity 



which agrees with von Helmholtz's result, when his notation \ (k l) 

 is written for C. 



There is apparently nothing self -contradictory in the more general 

 values of 0(r). The- form 0(V) = Cr + A-J-B/" 1 , here considered, is 

 notable for the case G = 1 ; as then the law of electrodynamic action 

 between two changing charges would be simply that of the inverse 

 square. 



Next we shall consider the propagation of the electric displace- 

 ment fgh. We have 



dt dxdt dx 



dt 2 47r dxdt 



with two similar equations in g and h. 



