with Special Reference to the Microscope. 103 



If this be 2R, we have 



I 



C dq = 4tt/j9. sin pU. . . . (67) 



i. 



From (67) we see that the image of a luminous line, all 

 parts of which radiate in the same phase, is independent of the 

 form of the aperture of the object-glass, being, for example, 

 the same for a circular aperture as for a rectangular aperture 

 of equal width. This case differs from that of a self-luminous 

 line, the images of which thrown by circular and rectangular 

 apertures are of different types *. 



The comparison of (67) with (20), applicable to a circular 

 aperture, leads to a theorem in BessePs functions. For, when 

 q is finite, 



r _ 2 2J 1 { N /Qg + g *)R} . 



so that, setting R=l, we get 



M^m dq= ^p (69)t 



y/(p* + q 2 ) 2 p 



The application to a grating, of which all parts radiate in 

 the same phase, proceeds as before. If, as in (58), we 

 suppose 



A(p) = Aq + .. .+ A r cob sp+ . . ., . . (70) 

 we have 



f+°° 

 A r = I d cos sp dp ; (71) 



from which we find that A r is 4-7T 2 or 0, according as the 

 ordinate is finite or not finite at x=s. The various spectra 

 enter and disappear under the same conditions as prevailed 

 when the object was a row of points ; but now they enter dis- 

 continuously and retain constant values, instead of varying 

 with the particular ordinate of the object-glass which cor- 

 responds to x=s. 



We will now consider the corresponding problems when 

 the illumination is such that each point of the row of points 

 or of the grating radiates independently. The integration 

 then relates to the intensity of the field as due to a single 

 source. 



By (9), (10), (11), the intensity I 2 at the point {p, q) of 

 the field, due to a single source whose geometrical image is 



*Enc. Brit, " Wave Theory," p. 434. 



t This may be verified by means of Neumann's formula (Gray & 

 Matthews, ' Bessel's Functions ' (70) p. 27). 



