aci) 



Curved Diffraction-gratings. 419 



wide ; it will require reducing in the ratio of 1 : x/19'2, or 

 about 3:8. 



To consider now the lateral Fig. 3. 



aberration, using the same 

 notation, describe a circle 

 (fig. 3) through Q, with A 

 as centre. Light from A q 

 arrives in the same phase at 

 all points on this circle. Let 

 Q' be the point on the circle 

 at which the light arriving 

 from P is in the same phase 6" 

 as that from A, and let 

 ty + hty be the angle OAQ', 

 and let i/r, 0, u, and v! satisfy 

 (5) and (6). Our funda- 

 mental equation (4) becomes, neglecting terms like co 2 $ty, 



sin <f>— sin yfr — Syfr cos yjr 



(o f , , /cos 2 6 cos 2 ilr\ 



- ¥ {cob*+ cos^-a(-^- + -^-) 



-sin^l- 2 ^)} 



-y|sin0^-%os0+^ 2 cos 2 </>) 



ini|r^l-^cos^+^- 2 cos 2 ^)j] = ±nX. (9) 



g^jcos^lsin^^- 2 ^^)} 



+ Y[ 8in *{i-i cos *(i-s COfl *)} 



-sint{i-^cos^r(l-^cos^)|] = 0; 



and, to the approximation adopted in considering the longitu- 

 dinal effect, 



+ ^sin^rcos^(l-^cos^)}. , . (10) 

 If, as before, Q coincide with 0, then u~0, <f> = 0, u' = a cos yjr; 



sm 



Hence 



