Thui 



oV= 



Curved Diffraction-gratings. 417 



To discuss, then, the aberration in this case. Let u\ = v! 4- 5V, 



where u' satisfies (6), and suppose we neglect oV| 2 , atuoV,and 



euch terms. Then 



( cos 2 6 . cos 2 ^ /., oV\ 1 

 cos $ + cos *-« I -jj- + -jp- {1 - v ) I 



+am { + ^- asin * c ° s * (l_£ C08< fr) 



i. "~ oao) w \ u / 



a sin -dr cos ijr / a , \ ) 



t ^- T ( 1 -^ cos t)j=o. 



—. "J ± q sin <f> cos 6(1 cos<f>) 



a cos y I* ° a( ° u \ u I 



+ ^sm^rcos + (l-^cos*)|. . . (7) 



Equation (7) determines the aberration in the general case. 

 To determine the effect of this in practice, let us suppose that 

 we are considering the spectrum of the first order, so that the 

 retardation of the light coming from two consecutive lines is 

 just one wave-length ; and hence, if P be on the #th line from 

 A, and a- the distance between two lines, then the arc AP = 

 k<r=ia(D, and n = k. 



Let us suppose, further, that the origin of light is at the 

 centre of curvature of the grating, so that u = a, <f> = 0, and 

 hence, taking the — ve sign in (6), 



sin yft= -, cos^=/y/ (l- ^ V u'=a co^=a^/ (l - \ 



Thus 



JcX 

 8M ' =+a<B 3^ = + * a - • • («) 



Let Qi (fig. 2) be the point on the line given by sin^=-> 



c 

 which is determined by w / =acos / ^r, Q' being the point on 



Fig. 2. 



that line at which light arrives in exactly the same phase from 

 A and P. Then QiQ'^&X. And the- angle PQ a A differs by 



