the Colour-Sequence in the Spectrum. 221 



consequently to be confined to finite limits — z 1 and -f^, 

 where 2z 1 is the width of the slit S. With the notation of 

 Lord R-avleigh, this gives 



1= f +21 ^-~ 2 sm 2 h x z[l + cos{p' + (*T-2h 1 )z}]dz. . (3) 



In the special case when w = 2/^ corresponding to the 

 best thickness of the retarding plate for observing Talbot's 

 bands, this reduces to 



I = (l + cos/o) | sin 2 Aj£^z. . . (4) 



When p = (2ji+1)7t, this will give zero illumination what- 

 ever z x may be. So that in this case we shall have Talbot's 

 bands with maximum visibility irrespective of the width of 

 the slit. 



But when the same plate is turned over to the position P 2 , 

 '37— —2h 1 and 



J+z x 9 

 — ^— 2 sin 2 /^ [1 + cos (p' - 4Ji x zJ\ dz. . 



(5) 



This integration can be easily performed with the aid of 

 the tables of the Si function given by Grlaisher *, for it can 

 be easily put in the form : — 



1 = 2 rSi2^ 1 - S,I ^ i ]-cos / o / [4Si4^-3Si6^ 



2 . -I 



— Si 2d'! H — smVi cos 4#j 

 #i J 



=A + Bcosp', (6) 



where m 1 -=hiZi. 



When Xi is very small, Si «z > 1 = ^ > 1 and 



I = 2[2x 1 -x 1 ]-cosp'[16 l v l -18<v 1 -2.v 1 + 2 t v l ] 

 == 2a?i(l + cos p'), 

 a value identical with that obtained from (4). 



* Phil. Trans. 1870, p. 367. 



