370 Dr. Barton and Mr. Lownds on Reflexion and 



We have therefore to determine theoretically the ratio y, of 

 the mean reading d' with the condenser on the line, to the 

 mean reading d with no condenser on. Now the electrometer- 

 throw, as previously explained, is proportional to the time- 

 integral of the square of the amplitude of the wave-train 

 passing it. Hence, denoting by E the electrometer-constant, 

 we have 



Ed = e-2< \ e-^Ptt-lkv) G0 ^p(t-y v )dt 



*■ IJv 



— *-2ol, 



: e-**?* co&pi dt, 



Jo 



or 



Mp(l + k*)e 2al >Fjd = 1 + 2P. . . . (20) 



32. Consider now the throw d' obtained with the con- 

 denser on the line. Retaining the previous notation for the 

 transmission and reflexion operators of the condensers we 

 have for the wave-trains successively transmitted at the 

 condenser the following series : — 



First wave-train : Be-kpt cos (pt -j- 0), 



Second do. CAB 6-^-2^ cos(pt + y + a + /?), 



Third do. C 2 A' J B*-^- 4/ *eos (/>*+ 2y + 2* + /3). ' 



The formation of this series is easily traced by noting that 

 each time a wave-train is incident upon the condenser, not 

 only is a part transmitted but also a part is reflected, suffers 

 attenuation along the line in its passage to the oscillator, is 

 there reflected, and again suffers attenuation along the line 

 before reaching the condenser a second time. These effects 

 are all provided for in the above series, in which also the 

 simplification is adopted of taking t — for each new arrival 

 of a wave-train at the condenser. 



33. We thus have, on taking the time-integrals of the 

 squares of the successive trains, the following series: — 



r* 00 



Ed'e^l* = B 2 e- 2 W cos 2 (pt + /3)dt 



s\ oo 



+ r 2 B 2 I e - a*P* cos 2 (pt + /3 + K)dt 



J° 



/loo 



+ r 4 B 2 1 e~ 2 W cos 2 (pt + /3 + 2/c)dt 



+ .+ + , • . . m) 



where r=UAe- 2l(T and /c=a + y. 



