with Special Reference to the Microscope. 191 



In the method referred to the form of the aperture is sup- 

 posed to remain symmetrical with respect to both axes, but 

 otherwise is kept open, the integration with respect to x 

 being postponed. Starting from (12) and considering only 

 those points of the image for which 77 and q in equation (8) 

 vanish, we have as applicable to the image of a single lumi- 

 nous source 



C = ft cospx dx dy = 2§y cospx dx . . . (57) 



in which 2y denotes the whole height of the aperture at the 

 point x. This gives the amplitude as a function of p. If 

 there be a row of luminous points, from which start radiations 

 in the same phase, we have an infinite series of terms, similar 

 to (57) and derived from it by the addition to p of positive 

 and negative integral multiples of a constant (py) repre- 

 senting the period. The sum of the series A (p) is necessarily 

 periodic, so that we may write 



A(^)=A + ... + A r cos(2?'7r/>/pi) + ... ; . (58) 



and, as in previous investigations, we may take 



r + 00 

 A r — I C cos sp dp, (59} 



J —00 



s (not quite the same as before) standing for 2r7r/p 1 , and a 

 constant factor being omitted. To ensure convergency we 

 will treat this as the limit of 



;: 



e ±hp cos sp dp. .... (60) 



the sign of the exponent being taken negative, and h being 

 ultimately made to vanish. Taking first the integration 

 with respect to p, we have 



f: 



±hp 7 h t h 



e v cos xp cos sp dp = 2 ^ + -^ 



and thus 



. P hy dx C hy dx 



Ar ~ J A 2 +V+*) 2 J h*+\x-s)*> 



in which h is to be made to vanish. In the limit the inte- 

 grals receive sensible contributions only from the neigh- 

 bourhoods of x= ±s; and since 



du ■ (61) 



. 



JL+M* ' 



P 2 



