from the Boundaries of Diffracting Apertures. 115 



appear on either side of the boundary of the circular 

 aperture with this arrangement. The luminosity, as in 

 the case of the simple knife-edge test, tends to zero at the 

 ends of a horizontal diameter. The explanation of this 

 fact and of the peculiar form of the fringes will appear 

 later. 



More striking still are the interference phenomena 

 obtained when the boundary is observed through a pair 

 of apertures of the same form placed in the focal plane. 

 With two horizontal slits placed on the same side of the 

 centre of the field, lunette-shaped interference fringes are 

 observed, the central fringe which coincides with the 

 boundary being white (PI. III. fig. 15). But when two 

 horizontal slits are placed on opposite sides of the centre 

 of the field, the central fringe is black — in other words, the 

 boundary of the aperture is itself non-luminous but appears 

 surrounded on either side by luminous bands (PL III. fig. 6). 

 As has been remarked in the introduction, this remarkable 

 fact is one of great generality. Fig. 16 in PI. III. represents 

 the appearance of the circular boundary when a horizontal 

 wire is placed across the centre of the focal plane. A fine 

 black line may be seen running through the luminous arcs 

 and dividing them into two. A case that admits of detailed 

 mathematical treatment is that in which the arrangement is 

 completely symmetrical about the axis. Figs. 7 & 10 in PI. III. 

 show the results obtained when the central part of the 

 field at the focal plane is blocked out by a circular disk and 

 only the diffracted rajs passing through an annulus of 

 greater or less width surrounding it enter the observing 

 telescope. It will be seen that in both photographs the 

 boundary appears as a perfectly black circle, with luminous 

 rings on either side of it. 



Let R be the radius of the circular aperture of the lens, 

 and assume that in the focal plane there is a screen con- 

 taining an annular aperture, E 1? R 2 being the radii of the 

 circles defining the annulus. Let <j> be the angle of 

 diffraction of parallel rays which meet at any point Q 

 in the focal plane. Since the path-difference between 

 the rays leaving a point (V, 6) and the centre of the 

 diffracting aperture is evidently r cos sin (/>, the diffracted 

 disturbance at a point in the focal plane due to an area 

 vdQdr can be written in the form 



. ft ?'cos(9siii(p\ m , 

 r sin 2tt I r j, - J dd dr. 



12 



