326 Mr. J. Bridge on the Diffraction of Light. 



(11) The result of summation for two points at distance a, axis 

 of X being taken parallel to the line joining the two, is that the 



intensity is 4cos^( -—j ). For fom* points in the corners of a 



squareit is 16cos^( — -r) cos^( — y j. For three equidistant 



points, oblique axes being taken perpendicular to the sides of 

 the triangle, it is 



3 + 2 cos^" "^ [x + y) + 2 cos— — r- x + 2 cos-— -^ y, 



or 



TTU v^3 , , 7r« v/3 TTfl -v/S 



1 + 8 cos -. (a; + 7/) X cos ^ .r x cos— — ^ y. 



The interpretation of these formulae is very simple, and will 

 be found to agree with the phsenomena, at least in their most 

 prominent characters. 



(12) The effect of a numerous series of points in regular order 

 may be best obtained from the consideration, that in any direc- 

 tion in which the retardations of the light differ for different 

 points by whole wave lengths, the effects of all will coincide, and 

 there will be a maximum. 



Take a series of points in equidistant parallel order, the rows 

 being a a' a" a'", bb'b"h"', &c., or across abed, a'b'c'd', &c. In di- 

 rections perpendicular to aa', ab', ac', ad', &c., by (5) and (3), 

 there will be maxima at distances inversely as the distances be- 

 tween the lines aa' , bb' ; ab', bd ; ac', bd', &c. Nov/ the pro- 

 duct of the distance of two consecutive points in any of these 

 rows by the perpendicular distance between it and the next row 

 is always the same, being the area of the smallest parallelogram 

 that can be formed by the points. From this it follows that 

 the arrangement of the maxima is similar to that of the aper- 

 tures turned through a right angle. 



(13) An aperture of the form of an ellipse gives an image 

 formed of a series of ellipses similar to itself turned through a 

 right angle. 



To explain this effect of an elliptic aperture, suppose all chords 

 perpendicular to a given line to be moved in the direction of 

 their length in such a manner that their middle points may lie in 

 the given line ; let them then be all altered in such a proportion 

 that the new ellipse may become a circle, whose radius is the 

 perpendicular from centre on a tangent parallel to the chords. 

 These changes, by (3), do not affect the alternations of light and 

 darkness in the focal line perpendicular to the chords. Now in 

 comparing the effects of two circular apertures by (5), corre- 



