210 Messrs. E. Edser and 0. P. Butler on a Simple 



is very approximately equal to half a wave, a may be taken 

 as wholly real. 



Consequently, if the thickness of the air-film is d, the re- 

 sultant transmitted beam on emergence will be given by the 

 real part of the sum of the infinite series 



.Mg^-YW. T -^— d (3) 



Substituting cos 2pd + is\n 2pd for e 2l P d , and rationalizing 

 the denominator of (3), we find the transmitted wave to be 

 equal to the real part of 



J^ e cp(x-Vt+d+e) 



where 

 and 



1 + a 4 — 2a 2 cos 2pd 



A'= , , , ".,.,..0-, , .... (4) 



sin 2pd 



tan pe = -z r, — '—= — ? 



1 \—a cos Ipd 



Now A 2 is proportional to the intensity of the transmitted 

 light ; hence as d is varied the intensitv will vary from 



IP V 2 



I(max.) = v 1 _ a . ) 2 to W) = (T^*f ' Takin g Michelson's 



expression for the visibility of interference bands, viz., 



W)-I(min.) we find that this becomeg equal tQ _1*L } 

 -•-(max.) + l(min.) 1 + tt* 



which will have a maximum value when a = l. Thus the 

 visibility of the bands will increase with the reflecting power, 

 and therefore with the thickness of the silver. 

 Also, from 4, we obtain 



dl_ -4pa*b* sin 2pd 

 dd~ C (l + a 4 - 2a* cos tpdf ' 



Consequently I varies much more rapidly as d is increased 

 when 2pd is nearly equal to 2nir than it does when 2pd is 

 nearly equal to (2n + l)ir. The bright bands will therefore 

 be very narrow and sharply defined, separated by broad 

 intervals very nearly black *. 



* A similar result has been noticed in connexion with the interference 

 of electrical waves. See " Electrical Interference Phenomena, somewhat 

 analogous to Newton's Rings, but exhibited by waves passing along 

 wires of which a part differs from the rest." By E. H. Barton, D.Sc, 

 Proc. Roy. Soc. vol. liv. p. 85. 



