548 THE ELECTROMAGNETIC FIELD. [PT. III. CH. XIII. 



Changing the order of summation and putting 



q=q'+p> 



_ =oo o' = oo / 1 \ p /2O' ,.2 



(38) /,^-2 *.* 



which by the definition of I p * is equal to 



p oo / _ T \p . .2jp 



(39) -- 



Differentiating by and adding to itself, 



which by the formula connecting the derivative of I p with the 

 contiguous function 



[Gray and Mathews (141)] may be written 



=- -i^ / 





, 



Inserting in the integral (35), and integrating from x to t, 

 (42 ) F- 



The terms free from a; may, by writing out the sums for I p , and 

 collecting terms, be shown to be equal to ff. 



We, therefore, finally obtain for V 



/ x 



(43) 



From this the values of F have been calculated and plotted by 

 Mr W. P. Boynton in Fig. 99, which shows the distribution of 

 potential along the line to the right, for times t = 1, 2, 3, 4, 5 times 



* From (27) we have for the present I p (f), 



