with Special Reference to the Microscope, 189 



disappears is given by, when we introduce the value of in 

 from (33), 



^«r ±r)=±1; 



or, since (as we have seen) — = — - — , (44) 



e (sin 7 + sin a) = + r\ (45) 



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. 2) 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 de- 

 duced from the formula (21) which gives the image of a 

 single luminous centre. 



If we limit ourselves to the case of parallel waves and 

 perpendicular incidence, the infinite series to be discussed is 



A ^ = J iW , JiO' + p ) j Ji("-tQ j J 1 (u + 2v) t ^ „ g ; 

 u u + v u — v u-\-2v 



where 



« = 7rf.2R/ty*. (47) 



Since A is necessarily periodic in period v, we may 

 assume 



A(w) = A + A, cos (2ttu/v) + . . . 4- A r cos (2r7ru/v) + . , , ; . (48) 



and, as in the case of the rectangular aperture, 



1 f + =° J^u) , . 2f +w J x (u) 2riru . 

 ,= - \ v y du, A r = - I ——' cos - - du. 



» -• u v J-«> u v 



(49) 



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

 a be positive *, 



jV-Wfc- -^k^- • • • ( 5 °) 



* Gray and Mathews' l Besfcefs Functions,' 1895, p. 72. 

 Phil. Mag. S. 5. Vol. 42. No. 255. Avg. 1896. P 



