184 Lord Rayleigh on the Theory of Optical Images, 



the values of the constants I , l l9 &c, are known. For these 

 we have as usual 



I, 



= l^I(u)du, I r = 2 ^°I(u)co S ^du; . (24) 



and it only remains to effect the integrations. To this end 

 we may observe that each term in the series (22) must in 

 reality make an equal contribution to I r . It will come to 

 the same thing whether, as indicated in (24), we integrate 

 the sum of the series from to v, or integrate a single term 

 of it, e. g. the first, from — go to + oo . We may therefore 

 take 



T lf +QO sin 2 w it . 



_ 2 f +a0 sin 2 



2irru 



, cos du. . . . (26) 



To evaluate (26) we have 



J+°° sin 2 wcossw , f +Q0 1 d , . 2 N , 

 du = \ - — (sin u cos su) du. 

 _oo u 2 J_oo udu y 



and 



d i • 2 \ s - 



-r (sin 2 u cos su) = — s sin sw 



+ — — sin (2 + s)w + — r-sin (2— s)u; 



so that by (15) (s being positive) 



,+Q0 sin 2 ^ cos su , f * . 2 + s , 2 — 5 



X 



d 





the rm'nus sign heing taken when 2 — 5 is negative. 

 Hence 



I,-£(l-=> «<>,.... (27) 



according as u exceeds or falls short of tit. 



We may now trace the effect of altering the value of v. 

 When v is large, a considerable number of terms in the 

 Fourier expansion (23) are of importance, and the discon- 

 tinuous character of the luminous grating or row of points is 

 fairly well represented in the image. As v diminishes, the 

 higher terms drop out in succession, until when v falls below 

 2 1 only I and I x remain. From this point onwards I x con- 



