The Abbe Diffraction Theory. By J. W. Gordon. 385 



the given aperture to this front is only '—^r *— . This, then, 



affords a measure of the illuminating effect of the wave-front segment 

 upon a point of space so situated with regard to it. Whereas it 

 would send to the focal point p an impulse proportioned to its whole 

 expanse, the impulse which the point now under discussion could 

 receive would be proportioned only to that which the given wave- 

 front could discharge in three parts of time out of 30| x 8 = 245 

 parts, these three parts of time being utilised only to the extent of a 

 discharge at the average rate of 7<S p. c. of the full discharging capacity 

 of the segment. Thus a point of space asymmetrically situated with 

 regard to the wave-segment, and so much over its geometrical shadow 

 that thirty and five-eighths sets of complete undulation phases concur 

 to reach it from the wave-front segment at every instant of time, 

 receives only 78 p. c. of 245, = nearly 1 p. c , of the light received by 

 the focal point. Now, the distance over the shadow boundary which 

 suffices to cause such an- amount of retardation as is here assumed, in 

 the discharge of the light impulse, is very small. The wave-length 

 of 50000 m - multiplied by 30§ amounts only to about the sixteen- 

 hundredth part of an inch. If we suppose that the wave-segment 

 has a rectangular form, a diameter of half an inch and a focal length 

 of ten inches ; it is easy to see that a displacement of no more 

 than g^ in. away from the focal point would plunge the observer into a 

 shadow where the amplitude of the light wave was reduced to this 

 small fraction (yJo) of the focal amplitude. If he goes farther 

 still away, the darkness will grow denser without sensible mitigation, 

 for the numerator of the fraction can never increase to more than 



2 sin ( j , and the denominator increases continually and indefinitely. 



2 sin - 7r 

 The fraction which works out at - ' ., n .-* — in the above example, 



8 



and represents the efficiency of a given wave-front, considered as a 

 source of light to a given point, may with advantage receive a name. 

 I propose in what follows to speak of it as the aperture- value from 

 that point of that wave-front. 



The case that has just been discussed is that of a point situated 

 well within the full shadow. The case which it is necessary to 

 consider in discussing the form of the antipoint is that of the 

 region lying close about the focus and in the shadow boundary. The 

 diagram already employed will serve for the investigation of this case, 

 if we assume the difference of the paths from p 2 to the nearer and the 

 further edge respectively to be much less than that already supposed, 

 say, for example, to be = A. — i.e. one wave-length. Then the wave- 

 front would of course occupy one wave-period of time in discharging 

 its energy on the point p 2 , and on the assumption of a uniform rate 

 of discharge during this period of time, the phase value of the wave- 

 front for the point p 2 would be - . In other words, the complete 



Aug. 21st, 1901 2 r> 



