CURRENT IN A WIRE OF FINITE LENGTH. 229 



The curve represented by this series would be very easily calcu- 

 lated, for the terms rapidly decrease when v differs appreciably from 

 unity. 



237. Sir W. Thomson has, however, solved the problem by the 

 help of another series, which is more easily discussed, and which 

 follows directly from Fourier's equation (4). 



According to this formula, the expression for the current at the 

 distance x from the origin, is 



1 ^ v v of vV= - t ** 1 * 

 1= --. -2 1 + 2 V e c^ 2 cos * . 



p ^x pl\_ - 1 / 



For the end of the wire which is in connection with the earth, 

 x = /, and we get 



Giving to n the successive values i, 2, 3 . . . , the cosine takes 

 alternately values equal to - i and + i. If for the sake of brevity 

 we put 



IT"* 



we get 



(12) I = l - 



For very small values of t, u tends toward unity, the series 

 in the parenthesis is equal to - , and the current null. As the 



time increases, u diminishes, the series tends to zero, and the 



y 



current diminishes up to a limiting value 7. 



pi 



The series can, moreover, be easily calculated; according to 

 Sir W. Thomson, it does not differ appreciably from its maximum 



value, until u is greater than - If a' is the time at which this 



4 

 value is attained, we have 



3 -^' " 2 / 2 7 /4 

 - = e a/, or a --rjr-/, ( - 



4 ^ \3 



