254] ELECTROMAGNETIC WAVES. 537 



If we now write u instead of u */KR/Z this is 



(13) F=C" 



Jo 



This definite integral is a function of its upper limit, and 

 therefore of x and t, satisfying equation (6). For x > and t 

 the value of the integral is VTT/S*. As we may add any constant 

 to V, we will put 



f 9 /t 



(14) F=F (l--^ 



\ NirJQ 



Thus for x > and t = 0, F= 0. For x = 0, t > 0, the value of 

 the definite integral is zero, so that V F . Consequently the 

 solution (14) represents the result of connecting one end of the 

 cable with a constant battery, and leaving it permanently con- 

 nected. 



The definite integral in (14) is the transcendent known as the 

 probability-integral, for which numerical tables have been cal- 

 culated. From these the values of V have been plotted, showing 

 the potential at the different points on the cable, Fig. 96, the 

 different curves being for times 1, 2, 3, 4, 5 times KR. It is to be 

 noticed that however small the interval of time from the instant of 

 connecting the battery, the disturbance is felt somewhat at all 

 points, however remote. Thus the velocity of propagation would 

 be infinite, if we could speak of a velocity. This shows that the 



This may be shown as follows. We have 



J: 



Consequently 



f 00 f 00 



Ixdy. 

 J o 



Changing to polar coordinates, 



Too T 



= e- x *dx = 

 Jo J 



= f 00 t" e-W+ 



Jo Jo 



fir*-! 



Therefore 



