362 Dr. Bartou and Mr. Lownds on Reflexion and 



valid to the head of the wave-trains, we have 

 <k + f = &=Q/S', 

 i^fa/Lv, i / =—^>'/Lv, ?' 2 = </> 2 /Lv, 



., ._dQ > ■ • • (1) 



Q = cf> 2 = for t = 0; J 



where Q is the quantity of electricity at time t on the positively 

 charged plate of the condenser of capacity S' (in electro- 

 magnetic units), and the i's denote the currents in the positive 

 wires due to the wave-trains denoted by the $'s bearing the 

 same subscript or dash ; L denotes the inductance of the line 

 per unit-length, and v is the speed of free aether radiation. 

 These equations were used by Heaviside with reference 

 primarily to waves of telephonic frequency, but are here 

 adopted as a working approximation for the phenomena which 

 obtain in the case of Hertzian waves. 

 On eliminating the i's and Q, we have 



(L^| + 2y=-L^|^, ... (2) 



(L*S'^+2)fc = 2fc . . (3) 



These expressions might have been written at once from 

 Heavisido's operational results, the reflexion and transmission 

 coefficients for an intermediate bridge being respectively 



-Lv , 2Z 



W^Lv and 2ZTT^' • • • • W 

 where the resistance operator Z is here equivalent to 



HT 



since the bridge is a condenser merely. 



16. Now if the incident wave-train is a damped sine 

 function right up to its head, we may write without loss of 

 generality, 



fa^e-W cos (pt + t) (5) 



Substituting this value of <f> x in equations (2) and (3), and 

 satisfying the initial conditions expressed in (1), we obtain in 

 the usual way the following solutions : — 



4>' = A*-Mcos (pt + t + a) — B cos (t-f /3)<?-V/r, . (6) 

 £,=?Ifc-*p'oos(p* + t + /9)-B cos (* + 0)g-%*/x' ; . (7) 



and 



