Curved Diffraction-gratings. 



415 



be placed in the focal plane of that lens, the screen would 

 practically be at an infinite distance from the grating but for 

 the aberration produced by the lens. 



So far, then, as definition merely is concerned, we have to 

 compare the aberration effects produced by these lenses with 

 those caused by the curvature of the grating. Of course a 

 reflexion gratia g used without lenses has an immense advan- 

 tage for experiments on the violet or ultra-violet rays which 

 are absorbed by glass. 



In considering the aberration, then, we shall follow the 

 method adopted by Lord Rayleigh in his paper on " Investi- 

 gations in Optics, with special reference to the Spectroscope. 

 Aberration of Lenses and Prisms " (Phil. Mag. January 1880). 

 Let QA, QP be two 

 adjacent rays diverging 

 from a point Q and fall- 

 ing on the concave side 

 of a circle A P, centre 

 0. Let QAO = </>, 

 AOP = co, QA = u, 

 OA = a. Then 



QAP = <£ + 



AP = 2asin^. 



ft) 



2' 



Hence 



QP 2 ==w 2 + 4a 2 sin 2 ^ — 4ausm~sin( ~— <j>); . . (1) 



and, expanding as far as a) 3 , we find 



QP = w + ao)sinc/> 5- ( cos0 cos 2 <£ J 



aw 3 sin 6/1 a a 2 A 



Again, let Q x be another point on the other side of the normal 

 OA, and let Q 1 A=u r , QiA0 = ^. Then 



Q 1 P = w / — a&> sin ifr ~l cosi/r 7 cos 2 ^ ) 



a&> 3 sim/r/l a a 2 2 . \ /ox 



+ — 2 V3~i7 cos ^ + ^" 2COS V * * ( 3 ) 

 Suppose now that A is a point on one line of the grating, and 



