526 The Electromagnetic Theory of Light [OH. xvin 



requisite extension of the theory. The reader who wishes to study the 

 matter more fully is referred to treatises on Physical Optics*. 



606. Our analysis has so far treated a current in a conductor as con- 

 tinuous in space. In point of fact we believe the current to be made up of 

 the separate currents produced by the motion of discrete electrons. The 

 motion of each electron will have associated with it a system of " displace- 

 ment" currents, and if the velocity of each electron is supposed small 

 compared with the velocity of light, then the whole system of currents will 

 be determined as soon as the velocities of the individual electrons are known 

 (cf. 585). 



Taking the displacement-currents into account, the whole system may be 

 supposed made up of a system of closed currents i 1} i z , .... The electro- 

 kinetic energy is, as usual, of the form 



T=t(L u i l *+2l l9 i l i t +...) (584). 



Let u lt v lt Wi\ u 2 , v a , w 2 ; etc., be the components of velocity of the 

 different electrons. Then each of the currents i l} i. 2) ... will be a linear 

 function ofu lt v lt w ly u 2 , v 2 , w 2 ,..., and the value of T given by equation (584) 

 will be a quadratic function of these quantities. Thus we must have 



T=^(a n u 1 2 + b u v l 2 +c ll w l 2 + ... + 2a M tt 1 w a +...) (585), 



where the coefficients a u , 6 U , a 12) ... depend on the structure and positions of 

 the electrons. 



In a metallic conductor we believe the moving electrons to be at distances 

 apart which are great compared with their linear dimensions. Let us con- 

 sider the extreme case in which the distances apart are infinite in comparison 

 with the linear dimensions of the electrons. In this case we may consider 

 that each electron is attended by its own system of displacement-currents, 

 and that there is no interaction between the systems associated with the 

 different electrons. Thus equation (585) assumes the limiting form 



T=i?ii(tt 1 +t; 1 + w 1 a + M a a +...) (586), 



since the coefficients of each of the terms u? t v^, w-f, u, . . . , must clearly 

 be all the same from symmetry. Clearly also the coefficient m, on the 

 hypothesis of electromagnetic mass ( 588), must be the mass of the electron. 



As the linear dimensions of the electron are imagined to decrease, the 

 intensity of the field at and near the surface of the electron will increase, so 

 that the quantity m will increase. If we consider the changes in the expres- 

 sion on the right-hand of equation (585) as the dimensions of the electrons 

 are imagined to decrease, while their distances apart remain unaltered, we 

 see that the coefficients a u , b n , ... will increase, while the coefficients a 12 , ... 



* For instance Drude's Optik, or Schuster's Theory of Optics. 



