224 Mr. J. Larmor. A Dynamical Theory [June 20, 



in which (a, &, c) is the velocity of flow of the sether where the electron 

 is sitnated, and is equal to the curl of (P, G, H) in such way that 

 the latter is Maxwell's vector potential given by the formulae of the 

 type 



F= |-dr+[lB T C r \- 



Jr J\ dz dyjr 



and where ^ is the electrostatic potential due to the electrons in the 

 field, so that ^r = c 3 2e/V, where c is the velocity of radiation. These 

 equations are proved to hold good, not merely if the motions of the 

 electrons are slow compared with radiation, as in the previous paper, 

 but quite irrespective of how nearly they approach that limiting 

 value; thus the phenomena of radiation itself are included in the 

 analysis. 



An element of volume of an unelectrified material medium contains 

 as many positive electrons as negative. This force (P, Q, B) tends 

 to produce electric separation in the element by moving them in oppo- 

 site directions, leading to an electric current iu the case of a conductor 

 whose electrons are in part free, and to electric polarization in the 

 case of a dielectric whose electrons are paired into polar molecules. 

 In the former case, the rate at which this force works on a current of 

 electrons (tt', v , w'), is Pu + Qv' + Bw/ ; it therefore is identical with 

 the electric force as ordinarily denned in the elementary theory of 

 steady currents. In the case of a dielectric it represents the ordinary 

 electric force producing polarization. So long as a current is pre- 

 vented from flowing, the ponderomotive force acting on the element 

 of volume of the medium is the one of electrostatic origin due to 

 such polarization as the element may possess, for as the element is 

 unelectrified it contains as many positive electrons as negative. But 

 if a current is flowing, the first two terms of (P, Q, B), instead of 

 cancelling for the positive and negative electrons, become additive, 

 as change of sign of the electron is accompanied by change of sign of 

 its velocity ; so that there is an electrodynamic force on the element of 

 volume, 



(X, Y, Z) = (v'cw'b, w'au'c, u'b v'a), 



where, however, (u ', v f , w') is the true current composed of moving 

 electrons, not the total circuital current (u, v, w) of Maxwell, which 

 includes the rotational displacement of the free rether in addition 

 to the drift of the electrons. 



The electric force (P, Q, B) as thus deduced agrees with the form 

 obtained originally by Maxwell* from the direct consideration 

 of his concrete model of the electric field, with idle wheels to 



* Maxwell, "On Physical Lines of Force," 'Phil. Mag.,' 1861-62; 'Collected 

 Papers,' vol. 1, pp. 450512. 



