8 Prof. A. Schuster's Electrical Notes. 



velocity. These equations differ from J. J. Thomson's by a 

 quantity which does not affect the magnetic forces, but which 

 does affect the energy if calculated according to equation (7). 

 It can be shown that equations (8) are the only ones which 

 satisfy the conditions of the problem, and give J. J. Thomson's 

 values for the magnetic forces. Owing to our ignorance of 

 what actually happens at the surface of the sphere when the 

 medium passes suddenly from a condition in which there is no 

 displacement to one in which there is a displacement, and vice 

 versa, the problem itself has a certain ambiguity ; and we 

 might possibly have to introduce some surface-currents at the 

 sphere itself. But these surface-currents would affect the 

 values of the magnetic forces as well as the vector-potential. 

 What I contend for is that equations (8) are the only ones 

 consistent with the values of the magnetic forces as defined 



b yW- . . . 



8. The energy of the field is now easily found with the help 

 of the following theorem : — Let a system of electric currents 

 be everywhere deducible from a current potential <£ so that 



0i= — -j- &c. The potential <fi is supposed to be continuous 

 dx 1 



throughout space and small, of the order ^, at an infinite dis- 

 tance ; but the differential coefficients of 9 may be discon- 

 tinuous at one or more surfaces. Let A 1? A 2 , A 3 represent 

 the components of a vector vanishing at an infinite distance 

 and satisfying everywhere the condition of no divergence, but 

 not necessarily continuous ; then 



jjj (A id + A 2 C 2 + A 3 C 3 ) dx dy dz = 0. 



To prove this it is only necessary to change the form of the 

 integral ; calling Z, m, n the direction-cosines of the normal 

 at the limits of the space to which the integral is applied, a 

 well-known transformation gives 



C[T/dA 1 </A 2 dA s \ , , , 



The second integral on the right-hand side vanishes every- 

 where, in consequence of the condition of no divergence, and 

 the surface integral when taken over both sides of all surfaces 



at which -~ is discontinuous will also vanish, provided $ 



itself is continuous. 



