CONTEMPORARY ADVANCES IN PHYSICS 315 



that is to say, whenever 



xo = mRyikw, y^ = 0, 2, 4, 6, . . . , (83) 



and attains its maximum value, double the amplitude of the uninter- 

 rupted waves, whenever 



Xo = mRyikir, k = 1, 3,5,7, ... . (84) 



Imagine circles drawn upon the plane of the screen, with their common 

 centre at the origin and their radii i?i, i?2, Rs, - ■ . prescribed by the 

 equations, 



mRk^lirXo = k, k = 0,1,2, 3,4, ... . (85) 



They divide up the plane of the screen into a tiny central circular area 

 and a series of surrounding rings. These are the "Fresnel zones" 

 relative to the point xo where the observer is placed. If the circular 

 hole comprises an odd number of the zones, the wave-motion at Xo 

 attains its maximum ; if an even number, the wave-motion vanishes — 

 there is silence or darkness. It seems as if the first, third, fifth and 

 other odd-numbered zones brought light, and the second, fourth and 

 other even-numbered zones destroyed it. 



It is equally easy to find the wave-motion along the axis of an an- 

 nular opening — that is to say, a circular hole partly filled by a con- 

 centric circular stop. Denote by i?o the radius of the -top and by R 

 the radius of the hole; then the limits of integration in (81) are super- 

 seded by p = Ro and p = R, and the amplitude along the a^is varies 



thus : 



A = ^211 - cos m{R' - Ro'')2xo]. (86) 



This contains the surprising conclusion that the maxima of amplitude 

 along the axis are as great as they would be if the stop were removed, 

 though they may be differently placed. An observer properly sta- 

 tioned should see the light brighten when the obstacle is inserted; it 

 may even be brighter than when the obstacle within the hole and the 

 wall surrounding it are totally removed, leaving no hindrance to the 

 onward march of the waves. 



The conclusion still holds good when the boundaries of the circular 

 hole retire to infinity, leaving nothing but an opaque disc in an other- 

 wise uninterrupted stream of plane parallel waves; although the ap- 

 proximations made in the foregoing pages are then no longer valid, and 

 equation (86) is not to be employed. Experience however shows that 

 when a small and accurately rounded circular disc is immersed in a 

 beam of parallel light there is a bright spot — more precisely speaking, 



