386 Transactions of the Society. 



series of phases which would arrive at every instant of time would 

 cancel one another, and at this point we should have a full shadow. 



The distance of the point p 2 from the focal point is evidently - d T 



(At 



where A. is the wave-length, d the distance from p 2 to the wave-front, 

 and a a function of the breadth of the aperture through which the 

 wave-front passes. This is the expression found by Sir Geo. Airy, as 

 above mentioned, for the radius of the dark ring defining the limits 

 of the luminous disc which represents a star in the telescope. In the 

 equivalent form of sin a . d, where a is the angle of diffraction, and d, 

 as before, the distance of the diffraction image from the aperture which 

 limits the wave-front, it is to be found in all the text-books as the 

 position of the first minimum in the series of diffraction spectra. It is 

 not necessary to repeat here a demonstration so easy to be found in 

 the text-books. The expression itself is only adduced because it very 

 readily yields the required definition of the shape and dimensions of 

 the autipoint. For it is to be observed that this function does not in 

 any way depend upon the amount of energy given out by the wave- 

 front ; that is to say, it does not depend at all upon the brightness of 

 the source of light. The counteracting forces are always accurately 

 balanced against one another ; what strengthens one must equally 

 strengthen the other, so that increasing the brightness does not even 

 tend to dissipate the darkness ; it may increase but cannot relieve the 

 strain. If we trace the line which forms the locus of this point, we 

 shall have a true geometrical boundary of the central disc of the 

 antipoint ; and a very little consideration will show what, in a general 

 sense, must be the distribution of light within that boundary. For 

 evidently all points situated nearer to the focns than this point must 

 confer upon the wave-front a finite aperture- value, for the fraction will 



be - , where m is the fraction representing the resultant of the 

 n 



series of wave-fronts that do not cancel one another, and n represents 



the fraction of a complete wave-period occupied by the discharge. 



At the focal point itself m = 1, for the successive wave-fronts do not 



overlap at all, and therefore no destructive interference can take place ; 



and n is evanescent, because no time is occupied by the discharge. 



Here the expression tells us only that the illumination is a maximum, 



not what its value is. But at every intermediate point both m and n 



must have finite values, and then the series of values — will indipate 



n 



not only the order of gradation, but also the steps in the gradation 



of the light. It is also evident that m will gradually fall off from 1 



to 0, and n will gradually increase from to 1. Thus there will be 



a continuous diminution of light from the maximum at the focal point 



itself to the minimum at the point p 2 . It does not, of course, follow 



that the light will be strong enough right down to this point to be 



perceived by the eye. On the contrary, it is clear that the visual 



