Electromagnetic Theory of Moving Charges. 491 



for circuits which lie partly inside the empty space enclosed 

 by the conductor. In particular, if there is a vector whose 

 line integral round every circuit in the field vanishes, the 

 lines of this vector must meet the surface at right angles. 

 Otherwise we should have a finite value for the integral round 

 a circuit drawn close to the surface outside and completed 

 inside. In other words, if a vector is derived from a potential 

 function, this function must be constant over the surface. In 

 the ordinary static case it is the electric force (X, Y, Z) 

 which is so derived ; but in the case of a steadily moving 



/ Z Y 



field it is the vector (X, Y, p J which meets the surface at 



right angles. ^ 



6. Let F(xy z)—C be the equation of the charged surface. 



Then <j>(xyz) has to be constant over this surface and satisfy 



dx 2 dy 2 dz 2 



Put 2=£f, then <j> is a function of x, y, ?, which is constant 

 when F(#, y,k£) = C, and which satisfies 



dfy.dtydty 

 dx 2 - dy 2 ^ dtp 



Therefore if we regard (x y f ) as Cartesian coordinates of a 

 point, <£> is the potential at external points of an electrostatic 

 free distribution on the surface F(#, y, k%) = C The com- 

 ponents of electric force due to this distribution, at a point 

 \x y f) on the surface, are 



dej) dej) d(f> 



dx dy d£' 



This force acts in the normal to the surface, and is pro- 

 portional to the surface-density at (x y J), which we shall 

 call <r f . Therefore 



/d<f> d<f> dcj>\ JdF d¥ dF\/ //rfFy , (dF\* TSFy 



But d _ , d 



dl~ di ; 



h erefore, denoting differentiation with respect to x y z by 

 ub scripts 1 2 3, 



(&, fe *•)= -A<7'(F 1; F 2 , F 3 )/ s/W+W+1^7. 



Now let o" be the surface-density at (x y z) on the moving 

 conductor F(# y z) = C, then equating a to the normal com- 



