Diffraction of an Annular Aperture. 9 



edly positive and negative, and the sum of all may be consi- 

 dered = 0. This reduces the expression for intensity to 

 E,2 + E,,S which will be easily discussed. 



Example 1.— The light enters through a round hole with- 

 out interruption. 



Here a = 0, and therefore e, = 0, and E^ = 1, and the 

 expression becomes 1 + E'f- 



The smallest value of this quantity occurs when E„ = 0. 

 This happens when e,, = 2-41, or = 5-52, or = 8-64, &c., or 

 when e,! is large. For all these cases the intensity is repre- 

 sented by 1 . 



The greatest value occurs when Ci, = 0, and the mtensity 

 is represented by 2 ; the next maximum is when e„ = 3-8, 

 and the intensity is 1-16 ; the third maximum is when e, = 7-0, 

 and the intensity is 1-09; the fourth maximum is when 

 e., = 10-], and the intensity is 1-06. 



Thus there will be at the centre a bright spot of double the 

 general intensity, surrounded by rings brighter than the great 

 expanse of light; but the excess of intensity in the rings, even 

 the first, is so small, that it probably could not be seen. The 

 whole diameter of the bright spot, to its very ext remities, is 



the visible diameter would be much smaller. 



Example 2.— The stream of light is unlimited externally, 

 but an opake circle of radius a interrupts a portion of it. 



Here e,, is very great, and E^,=0, and the expression for the 

 intensity of the light is simply E^. The intensities therefore 

 at different distances along any radius drawn from the central 

 normal parallel to the plane of the circle are represented by 

 the numbers in the last column of the table above. The first 

 maximum occurs when e, = 0, and it is then = 1, or is the 

 same as the intensity in an uninterrupted pencil ; the second 

 when e, = 3-8, and it is then i of the former ; the third when 

 e, = 7-0, and it is then j\ of the former; the fourth when 

 e^ = 10-1, and it is then j\ of the former. The light abso- 

 lutely disappears when e, = '2-4\, or = 5-52, or = S'M. Thus 

 there is a bright spot whose central intensity is equal to that 

 of uninterrupted light, surrounded by rings of much feebler 

 light, whose intensity decreases rapidly till it becomes insen- 

 sible. 



The whole diameter of the bright spot is 2 c, when e, or 



-JZ^L^ = 2-11, or c = ^:^.^*^L±Z X 2-41 ; or the dia- 

 \ V h^ + a^ 2 X a 



meter of the spot is __ x ^i 4-1. 



