190 Lord Rayleigh on the Theory of Optical Images, 

 Multiplying by bdb and integrating from to b, we find 



I 



J '< te >"&- «**P>—. . . (51) 



o x 



In this we write 5 = 1, a— is, where « is real. Thus 



r 



J/^){cos5^— zsin sx\ 



dx= s/(\—s 2 ) — is. 



If s 2 > 1 , we must write i V (,s 2 — 1) for V (1 — s 2 ) • Hence, 

 if 5<1, 



Ji (x) cos sx 



while, if 5>1, 



I 



I 



£ 



dx= V(l-^ 2 ), . . (52) 



J, (x) sins«r /KQ , 



-i-=-^ tf# = s; (5o) 



•I? 



— Li — dx = 0, (54:) 



J t (x) sin sx 



-^—- <fo=-vV-l) + *. • (55) 



We are here concerned only with (52), (54), and we con- 

 clude that A = 2/e, and that 



. 4 V(l — s-) A /K ,. N 



A r = — — -, or 0, ... (56) 



v 



according as 5 is less or greater than 1, viz. according as 

 2?'7r is less or greater than v. 



If we compare this result with the corresponding one (30) 

 for a rectangular aperture of equal width (2R = a), we see 

 that the various terms representing the several spectra enter 

 or disappear at the same time ; but there is one important 

 difference to be noted. In the case of the rectangular aper- 

 ture the spectra enter suddenly and with their full effect, 

 whereas in the present case there is no such discontinuity, 

 the effect of a spectrum which has just entered being infi- 

 nitely small. As will appear more clearly by another method 

 of investigation, the discontinuity has its origin in the sudden 

 rise of the ordinate of the rectangular aperture from zero to 

 its full value. 



