388 Transactions of the Society. 



this general resemblance will be so close that it would be scarcely- 

 inaccurate to describe the antipoint as a reproduction of the aperture 

 on a reduced scale, and with its axis turned through a right angle. 

 The mere geometry of these relations between aperture and antipoint 

 must not detain us now. Fig. 90 gives the figures of the antipoint 

 produced in the two important cases of a circular and a long rect- 

 angular aperture, moderate brightness of illumination being presumed. 



Fig. 90' 



Small figures have been inscribed in the apertures of the diagram to 

 show to the eye at once the forms and orientation of these antipoints, 

 and this general reference to the theory of these figures must suffice 

 for the present. 



It has been already seen that the expression - d gives the posi- 



a 



tion of minimum illumination, because the time occupied by the wave- 

 front in discharging its energy upon the point so determined gives 

 opportunity for the coincidence at that point of one complete set of 

 wave-fronts — i.e. of one set of wave-fronts in every possible phase of 

 vibration, and of no more than go to one complete set. It is obvious 

 that two complete sets would cancel one another in like manner ; so 

 also three complete sets, or any number of complete sets, if they 



coincided both in time and space. Consequently the point — d, 



3\ a 



— d, and so on, are also points of total darkness, with maxima rising 



to various intensities of illumination lying between them. That is to 

 say, the aperture-value of the given wave-front at any of these points 



situated in the same plane as the focus, and distant from it by - 



a 



(A, ; 2 A, ; 3 A ; etc.), is zero. The aperture-value at the interposed 



points of maximum illumination will evidently run down in a regular 



scale. Between - d and — d it will have the value ,-r— ; between 

 a a S 7r 



— d and ' d it will have the value _ : and so on ; where A 

 a a o 7T 



is a function of the diameter the precise value of which depends upon 



the form of the aperture. Though slightly overstating the degree of 



