184 Mr. W. Baily on the Spectra formed 



These equations give us 



a (sin $' + sin 6) — n{k ly 



cr (sin f — sin 0) = n 2 \ 2 . 



Now (fig. 2) let D be the centre of a cylindrical grating 

 whose lines are perpendicular to the plane of the paper, C the 

 centre of curvature, and CD = c. Let P be the source of 

 light, Q the focus of a diffracted ray, E a point on the grating 

 near to D. Join PD, PE, QD, QE, CE. Let CP=a, 

 CQ = b, ZPCD=a, QCD=/3; DP=r, DQ=/, ZCDP=0, 

 ZCDQ = 0', ZDCE = y. Then we have for light diffracted 

 back from the grating, which we may call " reflected light," 



sinCEP+sinCEQ = n-, 



a sin (a— 7) &sin(/3— 7) _ n\ 



{a 2 + c 2 -2accos(u-ry)}i + {b 2 + c 2 -2bccos(^-y)}^~ ^' 



Expanding in terms of 7, we get 



a sin a b sin (3 n\ 



1 + 



+ 



{a 2 + c 2 -2accosu}i {b 2 + c 2 -2bc cos /3}i <r 

 ■(a 2 + c 2 — 2ac cos u)a cos a — a 2 sin 2 a 

 (a 2 + c 2 — 2ac cos a)^ 



(P + c 2 - 2Z>c cos /3)cos /3 - 6 2 sin 2 /3 



> 



(b 2 + c 2 - 2bc cos £)* 

 + terms involving higher powers of y = 0. 



Putting 



acosa = c— rcos# 3 &cos/3 = c— /cos 0', 



asina=csin0, 6 sin /3 = c sin #', 



we get 



. „ .' \ l~COS0 cos 2 , cos0' cos 2 0'~l , B 



sm^+sm^-n- + + r — C7 + &c. = 



o- L c f c / J 



This equation must be satisfied for all very small values of 7. 



Hence . n \ 



sin t/+smc/'= — ; 



and cos 2 cos 1 



+ -7* 



r c d 



cos 2 <9' _ cosfl' _l 



/ c <i 



where cZ is any quantity. 



