430 Mr. 0. Heaviside on the 



C^-ti-iv/p^feivdz.eP*, . . (172) 



where C is given in (170). The summation here, with t=0 } 

 is therefore the expansion of C . 



The internal state of the wire is to be got by multiplying 

 the first w by such a function of r, distance from the axis, and 

 of whatever other variables may be necessary, as satisfies the 

 conditions relating to inward propagation of magnetic force, 

 and whose value at the boundary is unity. In the simple 

 case of a round solid wire, (172) becomes, by (87), Part II., 



Cr=Go ^L^l w J^±^ , . (178) 



This gives C r the current through the circle of radius r, 

 less than a x the radius of the wire, CV being the final value. 

 The value of s x is ( — Airfji^p^. Here of course we give to 

 //-!, ki, and <% their proper values for the particular value of z. 

 As before remarked, they must only vary slowly along z. 



In the case of a wire of elliptical section it is naturally 

 suggested that the closed curves taking the place of the 

 concentric circles defined by r= constant in (173) are also 

 ellipses ; and that in a wire of square section they vary 

 between the square at the boundary and the circle at the 

 axis. The propagation of current into a wire of rectangular 

 section, to be considered later, may easily be investigated by 

 means of Fourier series, at least when the return current 

 closely envelops it. 



As an explicit example of the previous, let us, to avoid 

 introducing new functions, choose the electrical data so that 

 the current-functions X and Y are the J and K functions. 

 This can be done by letting TL" be proportional and S" 

 inversely proportional to the distance from one end of the 

 line. Let there be no leakage, and 



W = TL % S=S z->; 



where S is a constant, and R " a function of d/dt, but not of z. 

 The electromagnetic and electrostatic time-constants do not 

 vary from one part of the line to another. The equation of 

 the current-function is 



J!(*S)= E °" S ^ • • • < 152 *) 



