468 Transactions of the Society. 



This is the condition, according to elementary theory, in order 

 that the rays forming the spectrum of the rth order should be 

 inclined at the angle a, and so (fig. 118) be adjusted to travel from 

 A to B, through the edge of the lens L. 



The discussion of the theory of a rectangular aperture may 

 here close. This case has the advantage that the calculation is 

 the same whether the object be a row of points or a grating. A 

 parallel treatment of other forms of aperture, e.g. the circular 

 form, is not only limited to the first alternative, but applies there 

 only to those points of the field which lie upon the line joining 

 the geometrical images of the luminous points. Although the 

 advantage lies with a more general method of investigation to be 

 given presently, it may be well to consider the theory of a circular 

 aperture as specially deduced from the formula (21) which gives 

 the image of a single luminous centre. 



If we limit ourselves to the case of parallel waves and per- 

 pendicular incidence, the infinite series to be discussed is 



Afa) - "£&U Jl(U+V) + Jl(U ~^ + J '(" + 2 *> +, (46) 



where 



w = tt£.2K/\/. .... (47) 



Since A is necessarily periodic in period v, we may assume 

 A (u)= A + A x cos (2mt/v) + . . . + A r cos {2rirtijv) + .<..; (48) 

 and, as in the case of the rectangular aperture, 



1 p - iW d A m a r* < JiW cos 2_r*n rfM 



These integrals may be evaluated. If a and b be real, and a be 

 positive,* 



$\'~ m3,mdx ~ j (* + *> •• ■ (50) 



Multiplying by b db and integrating from to b, we find 



'its: * , .■/(-+>■>-«. . ,„, 



r. 



o « b 



In this we write b = 1, a = is, where s is real. Thus 

 f*°° J\ (x) {cos sx — i sin sx) 



dx = ,J (1 — s 2 ) — is. 

 x 



* Gray and Mathews' ' Bessel's Functions,' 1S95, p. 72. 



