with Special Reference to the Microscope. 181 



But before proceeding to such calculations we may deduce 

 by Lagrange's theorem the interval e in the original object 

 corresponding to that between w = and u = ir in the image, 

 and thence effect a comparison with a grating by means of 

 Abbe's theory. The linear dimension (f ) of the image cor- 

 responding to u = 7r is given by j- = \f/a ; and from 

 Lagrange's theorem 



e/g = sin /3/sin a, .... (17 a) 



in which a is the " semi-angular aperture," and j3=a/2f. 

 Thus, corresponding to u = tt, 



€= iV sin a. 



The case of a double point or line represented in fig. 4 

 lies therefore at the extreme limit of resolution for a grating 

 in which the period is the interval between the double points. 

 And if the incidence of the light upon the grating were 

 limited to be perpendicular, the period would have to be 

 doubled before the grating could show any structure. 



When the aperture is circular, of radius R, the diffraction 

 pattern is symmetrical about the geometrical image (p = 0, 

 q = 0), and it suffices to consider points situated upon the 

 axis of f for which 7] (and q) vanish. Thus from (12) 



= 1 1 cos pes das dy = 2 I cospx\/(R 2 —w 2 ) dx . (18) 



This integral is the Bessel function of order unity, de- 

 finable by 



z pr 



J x (z) = — | cos (z cos <f>) sin 2 $ dcp. . . (19) 



Thus, if „v=R cos <£, 



C = ^ 2 -J«, (20) 



or, if we write u = w^ . 2R/\/, 



C = 7rR 22 ^H (21)* 



This notation agrees with that employed for the rectangular 

 aperture if we consider that 2R corresponds with a. 



The illumination at various parts of the image of a double 

 point may be investigated as before, especially if we limit 

 ourselves to points which lie upon the line joining the two 



* Enc. Brit, " Wave Theory," p. 432. 



