6 Prof. A. Schuster's Electrical Sotes. 



eicll, and it is the only one really consistent with the third 

 law of motion. 



7. The energy of the magnetic field established by the 

 moving sphere can be calculated either as Heaviside has 

 done directly from the volume-integral of the square of 

 the magnetic force, or as J. J. Thomson has done from the 

 vector-potential according to Maxwell's equation 



i\\\(A 1 C 1 + AX, + A,C,)<te clydz. ... (7) 



A is the vector-potential, G the current-density, and the 

 indices represent the components along the three axes. To 

 calculate the vector-potential for the system of currents which 

 are assumed to take place owing to the varying displacement 

 in the space surrounding the moving sphere, I consider first 

 the simple case of a sphere of radius a, the surface of which acts 

 like a sink of an incompressible fluid, the total quantity 

 abstracted in unit time being q. The current-density at a 

 distance r from the origin would be q/Airr 3 , and the current 

 components would be 



q aTc\~^Fry q d^\^r) q Tz\^r)' 



Inside the sphere there is no current. Each component of 

 the vector-potential, say A l3 has to satisfy the following 

 conditions : — 



V 2 A 1== -/^- if r >a, 



V 2 A 1 = if r<a. 



Further, A and its first derivatives are to be continuous at the 

 surface. 



Assuming the relation 



<j> n being a solid harmonic of degree n, the first condition is 

 satisfied by 



. __ qp , d 1 _ p.qx m 

 2 dx r 2r 



and to this we must add a solution of V 2 A = which, together 

 with the value just found, will satisfy the other conditions. I 

 obtain in this way, outside the spherical surface 



a _9^ (a 2 \ 



