460 Transactions of the Society. 



from calculations conducted by the same method, that a grating of 

 an equal degree of closeness would show no structure at all but 

 would present a uniformly illuminated field. 



But before proceeding to such calculations we may deduce by 

 Lagrange's theorem the interval e in the original object correspond- 

 ing to that between u = and u = tt in the image, and thence 

 effect a comparison with a grating by means of Abbe's theory. 

 The linear dimensions (If) of the image corresponding to u = tt is 

 given by f = Xf/a ; and from Lagrange's theorem 



e/f = sin /S/sin a, .... (17a) 



in which a is the " semi-angular aperture," and /3 = a/2f. Thus 

 corresponding to u = it, 



e = ^X/sin a. 



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

 therefore at the extreme limit of resolution for a gratins 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 perpen- 

 dicular, the period would have to be doubled before the grating 

 could show any structure. 



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

 tern is symmetrical about the geometrical image {p = 0, q = 0), 

 and it suffices to consider points situated upon the axis of f for 

 which tj (and q) vanish. Thus from (12) 



r +n 



C = // cos px dx dy = 2 cos^j.vj -y/ (R 2 - x 2 ) dx. (18) 



J - R 



This integral is the Bessel function of order unity, definable by 

 Ji (~) = - cos (: cos </>) sin 2 <f) d<f>. . . (10) 



Thus, if x = R cos (p, 



C = 7rft 2 2Jl <g B > , .... (20) 

 pR v 



or, if we write n = ir £.2 R/A./, 



C = 7rR 22Jl( ^ (21)* 



u 



This notation agrees with that employed for the rectangular aper- 

 ture if we consider that 2 It corresponds with a. 



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



