1893.] Phenomena somewhat analogous to Newton 9 s Rings. 91 



(Similarly a/a 2 = ft/ft z will be used for the value v 2 .) 

 It will thus be seen that the damping just referred to in the head- 

 ing of this paragraph is the primary damping, that is, the time-rate 

 of decrease of the oscillations occurring in the Hertzian primary 

 conductor. Secondary damping, on the other hand, refers to the 

 space-rate of decrease of any individual wave as it proceeds along the 

 Hertzian secondary conductor. It is, of course, along the long form 

 of this secondary conductor that we are now supposing the waves to 

 travel, but the secondary damping is known to be small in comparison 

 with the primary damping, and is, therefore, in the present part of 

 the theory, legitimately neglected.* 



16. It will readily be seen that the ordinary mathematical treat- 

 ment of the interference of light in thin plates will not strictly apply 

 to this case. 



For, in the optical phenomenon, one supposes a continuous beam of 

 light of constant amplitude. We may, therefore, in that case, at 

 once take, to infinity, the sums of the series of reflected and trans- 

 mitted rays to which the original one gives rise, and neglect the com- 

 paratively small period which elapses before those tivo infinite series are 

 made up, and during which (the series being as yet incomplete) the 

 reflected and transmitted beams have not reached their final steady 

 values. 



But with such primary damping as that with which we have to do 

 (namely, of the order ^ = 2 tra/p = 0'5), the character of the result 

 would be essentially changed by the unwarrantable assumption that 

 the amplitude of the incident wave remains sensibly constant until 

 the infinite series of internal reflections has taken place. 



17. The question is therefore attacked by forming a series of 

 integrals. 



Referring to fig. 1 or fig. 2, the wave given by equation (15) 

 advances in the positive direction along the axis of x, that is, in the 

 direction ABCD. Let t = when the head of the wave first reaches 

 C. And let x = for the wave which is at the point C, without 

 having suffered any internal reflections in the part BC. Thus, for a 

 wave which has suffered 2 ra internal reflections within the part BC, the 

 point C has for its abscissa 2nX I, where I denotes the length BC. 



Now let y n denote a wave emerging at C after 2 n internal reflections 

 in BC ; then we have, by putting, in equation (15), x = 2 nl, and 

 supplying the amplitude from (14), 



y n = ab zn (1 6 2 ) e 

 r y n = ab zn (l-6 2 ) e - <*-'> sin [ft (*-fe)], > (16). 



where t z = 2ljv z J 



* Compare ' Wiedemann's Annalen,' vol. 44, pp. 83 and 515, 1891. 



