I 



Transmission of Electric Waves along Wires. 369 



small (say one-millionth, of a second), that we may regard the 

 spark-gap of the primary as hot enough to act as a short 

 circuit to the secondary of the induction-coil, whose induct- 

 ance is of the order 20 henries. 



29. Electrometer-throw for Single Passage of Wave-train. — 

 It might at first be supposed that the phase at the head of a 

 wave-train would have no effect on the electrometer-throw, 

 except when that wave-train was interfering with another, 

 and so producing stationary waves in the neighbourhood of 

 the electrometer. This, however, is not the case. On the 

 contrary, the throw produced by the passage of a single 

 wave-train of given amplitude, period, and damping depends 

 to some extent on the initial phase. Thus we have 



r 7 , / n-,.,7 1 — &sin 2t + P(l + cos 2i) ,.„. 

 [e-W cos (pt + t)Ydt= 4 jfcp(i + #) l ' ( 17 ) 



Hence, although we have hitherto been compelled, in one 

 or two cases, to sacrifice rigour on minor points on account 

 of insufficient data, the initial phase of a wave-train will 

 hereafter always be taken account of when dealing with 

 reflexions for which theory furnishes the value of the phase- 

 change to be expected. 



It may be seen from (17) that where k = 0'l the value of 

 the integral varies with t between limits having the ratio 

 11 to 9 approximately. 



30. Transmitted System. — Let the condenser under exam- 

 ination be at a point on the line distant I cm. from its 

 beginning at the oscillator, and let the electrometer be l 2 cm. 

 beyond the condenser, i.e. l + l 2 from the beginning of the 

 line. Take the origin of coordinates at the condenser, and 

 let the potential-difference of the wires due to the incident 

 wave-train be represented by fa, where 



(j )l z=e-^P(t-^M-ffa: C0S p^t — x/v). . . (18) 



The a here provides for the attenuation of the waves in 

 their passage along the line. Its value has been previously 

 determined experimentally. 



The train incident at the condenser is accordingly 



fa=e-*pt cos pt (19) 



31. Now, owing to the irregularity of the sparks at the 

 primary oscillator, it is necessary in the experimental exami- 

 nation of any condenser to alternate electrometer-readings 

 with and without the condenser on the line ; the bridge 

 which absorbs all waves incident upon it being throughout in 

 place at the end of the line. 



