70 Mr. E. H. Barton. Electrical Interference Phenomena 



(1.) For the single reflexion at the beginning of the abnormal part. 

 Let the amplitude of the original wave-train be a, and that of the 

 reflected one be ab, then 



b - m 



r + 1 ' 



where r is the ratio of the electrostatic capacity of unit length of the 

 abnormal part of the secondary to that of its normal parts. 



(2.) For the interference of the ware-trains proceeding from the 

 beginning and end of the abnormal part of the secondary. 



Let the incident wave-train be 



<j> = oe-'+i sin (fit fax) (2), 



where denotes electrostatic potential, a/ctj = ft/ fa = v h the velocity 

 of propagation of the waves, and a, a and ft are constants depending 

 respectively upon the amplitude, the primary damping and the fre- 

 quency. 



Let the energy of the wave-train, when incident upon the beginning 

 of the abnormal part (BB' fig. 1) be t, let IT be the total energy of all 

 waves transmitted through the abnormal part, and let tp be that of 

 all the waves returned along BA after one or more reflexions at C. 

 Then 



26* 



~l + i 

 where 



-< O/fi) sin ffii + cos ptj- 



1 + 6 2 1 2 &*-"' cos 



and t-i denotes the time occupied by the waves in twice traversing 

 the length, BC, of the abnormal part of the secondary. 



APPROXIMATE THEORY OF DISTURBANCES. 



Thus far, then, we have dealt only with a single incidence of the 

 wave-train upon the abnormal part of the secondary, and have found 

 expressions for the two trains thereby produced. But in order to 

 represent more completely what occurs with the apparatus arranged 

 as in fig. 1, it is further necessary to trace the history of the two 

 wave-trains to which the original one thus gives rise. 



If the reflected wave-train were quickly extinguished and never 

 again reached the plates, SS' fig 1, and if the transmitted wave-train 

 after passing the electrometer, and thus giving the deflexion by which 

 it is measured, were also quickly extinguished without again reach- 



