42 2 On Curved Diffraction-gratings. 



or about 7*6 centim. A lens of this aperture would enable us 

 to use with the best advantage the whole of the grating if it 

 were plane ; whereas in the concave grating, for good defini- 

 tion we should only use about three eighths of the whole. It 

 would seem, then, that in cases in which there is no objection 

 to the use of glass (because of its absorbing qualities), a large 

 grating may be used to greater advantage if it be ruled on a 

 flat surface and properly chosen lenses be employed with it, 

 than if the grating be curved. 



It may be instructive to consider the subject briefly in an- 

 other manner. Let Q, Q l (fig. 5) be any two points, and 



Fig. 5. 



with Q and Q x as foci describe a series of confocal ellipses; let 

 the major axes of these ellipses increase in arithmetical progres- 

 sion, and let the common difference be A,. Consider a sphe- 

 rical wave diverging from Q and reflected at any point of any 

 one of these ellipses ; all the reflected light will reach Q x in the 

 same phase. Take any surface P 1? P 2 , &c. cutting the ellipses 

 in P 1? P 2 &c, and suppose it capable of reflecting light at these 

 points and incapable of so doing elsewhere. All the light from 

 Q which falls on this surface at these points will be reflected 

 to Q 1; and the various waves will reach Q 1 in the same phase. 

 If now Q be the section of a slit normal to the paper, Pj P 2 

 &c. that of a polished cylindrical surface whose generators are 

 normal to the paper, and lines be ruled on this surface to block 

 out the spaces F 1 P 2 , P 2 P 3 , &c, the lines also being normal to 

 the paper, we shall obtain a diffraction-grating which will give 

 an image of Q without aberration at Q x . 



We can thus determine the law according to which lines 

 must be ruled on any cylindrical surface to give an aplanatic 

 diffraction-image of a slit ; for we require only to write down 

 the equations to the ellipses and the surface and determine the 

 points of intersection. We will solve the simple case when 

 the curve T 1 P 2 &c. is a straight line parallel to Q Q x . Take 

 Q Qi as axis of x ; let a and b be the semi-axes of one of the 

 ellipses, suppose that which touches the line P x P 2 . . . ; then 



