31G MR, S. H. BURBURY ON THE INDUCTION OP ELECTRIC 



Again, any given phase of the disturbance occurs at a later time in the inner than 

 in the outer field, the difference of time being dt = a/X. In the case we are now 

 treating a = X/K, and a/X = I/K = Q/arh. The ratio of h to this time, or cr/Q, is the 

 velocity with which the disturbance penetrates the shell. Since it is independent of X, 

 it must be the same for all systems of currents of the type <f> on the surface S. 



We have thus obtained an answer to the question suggested in (17), so far as 

 regards a self-inductive system of currents. The velocity, namely, with which they 

 initially penetrate a solid, or, which is the same thing, the thickness of the stratum 

 which they may be supposed to occupy at a very short time after the commencement of 

 the induction, is proportioned to the thickness of the self-inductive shell at any point. 



32. The energy dissipated in the shell per unit of time is 2cT. 



We see then that, comparing similar self-inductive systems with different values of 

 K, but the same mean energy, the heat generated on average per unit of time varies 

 as K, or, as this heat must all be drawn from the batteries of the primary system, the 

 cost of maintenance of the system varies as K. 



Examples of Self-inductive Systems. 



33. A spherical current sheet. (See the works cited above.) 



Every spherical current sheet is self-inductive with ar/h constant, if <f> be a spherical 

 surface harmonic of any one order as A,Y M . For, the sheet being spherical, i/ = ; 

 and, by a known property of the sphere, 



,, 4?ra TT F ' 4rra 



r = L or = 



2n + 1 U 2n + 1 



when a is the radius. 

 Similarly, 



G 4? H 



Y ftt+1' W" 2n + l' 







The condition for (f> is then satisfied; and, as 47ra/(2n + 1) is constant, cr/h has 

 constant value over the surface, or the shell, if of uniform material, must in order that 

 the system may be self-inductive be of uniform thickness. In this case Q = 4vra/(2n + 1 ) 



and K = - - , if h be the uniform thickness of the shell. 



A h 



34. If S be a solid of revolution about the axis of z then any system of currents 

 on it, determined by an arbitrary function of z as current function, satisfies the 

 conditions F/U = G/V at every point with y = 0, H = 0, provided d<J>/dz be of the 

 same sign throughout S ; and, therefore, any such system may be made self-inductive 

 by suitably choosing a-/h. 



For the lines of resultant current are circles round points in the axis as centres, 

 and the lines of vector potential are also circles round points in the axis. Therefore 



