The Theory of Optical Images. By Lord Rayleigh. 469 

 If s 2 > 1, we must write i *J (s 2 — 1) for ,J (1 — s 2 ). Hence 



/: 



J iMH£ 4 = i; (53) 



while, if s > 1, 



fJiWWj;,., (64) 



j. 



a; 

 Jj (a?) sin «c 



o x 



dx= - ^/(s 2 -!) + s. (55) 



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

 that A = 2/v, and that 



A r = 4 ^- s2) , orO, . , (56) 



according as s is less or greater than 1, viz. according as 2 r it is 

 less or greater than v. 



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

 a rectangular aperture of equal width (2 E = 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 aperture 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 infinitely 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 aper- 

 ture from zero to its full value. 



In the method referred to the form of the aperture is supposed 

 to remain symmetrical with respect to both axes, but otherwise is 

 kept open, the integration with respect to x beiDg postponed. 

 Starting from (12) and considering only those points of the image 

 for which rj and q in equation (8) vanish, we have as applicable 

 to the image of a single luminous source 



C = JJ cospx dxdy = 2 fy cospx dx (57) 



in which 2 y denotes the whole height of the aperture at the 

 point x. This gives the amplitude as a function of p. If there 

 be a row of luminous points, from which start radiations in the 

 same phase, we have an infinite series of terms, similar to (57) 

 and derived from it by the addition to p of positive and negative 

 integral multiples of a constant (p x ) representing the period. 



