Curved Diffraction-gratings. 421 



opposite in phase to some of that which reaches Q t from be- 

 tween P 2 and A, and tends to neutralize the effect of this; while 

 similar results hold for light coming respectively from below P 2 

 and between A and P 3 . Thus a large aperture does not ne- 

 cessarily mean that there is a large quantity of light at the 

 focus. Exactly the same may happen in the case of a lens. 

 Lord Rayleigh has shown that if « be the angular semi-aperture 

 of the lens as viewed from the focus, and the curvatures of the 

 lens be adjusted to reduce the longitudinal aberration to a 

 minimum, « 4 should not exceed X/f. A similar course of rea- 

 soning shows us that if « 4 is greater than 2\/f y the light from 

 the outer annulus of the lens will be opposite in phase to that 

 from the central portions. 



To compare, finally, the definition of the curved grating 

 with that produced by a plane grating, and two lenses of equal 

 focal length used as a collimator, and the object-glass of a 

 telescope respectively, we can show (Parkinson, ' Optics/ 

 § 130), that if a. is the semi angular aperture of either of these 

 lenses seen from its principal focus, / its focal length, and the 

 curvatures are adjusted to make the aberration of each lens a 

 minimum, then the aberration is, for light of refractive index 



15 

 1*5, ~foc 2 ; but, as quoted above, Lord Rayleigh has shown 



that the aberration should not be greater than \/o£. Hence a 4 



7 A, 

 must not be greater than ^e 7 ' 



In the case of the first of Prof. Rowland's gratings discussed 

 above, the slit and eyepiece are at a distance of about 200 cen- 

 tiin. from the grating. Let us suppose we are using two lenses 

 of 200 centim. focal length, and inquire what their aperture 

 may be to allow the condition above given to be satisfied. If 

 y be the radius of the lens, we have 



. ,_ 7 x8xl0 6 x6 

 if not > than 15 x 1Q5 



Thus y must not be greater than 3*8 centim. A lens of this 

 aperture would just about admit the light from the whole of 

 the actual grating 5 centim. x 7 centim. in area if it were 

 plane; whereas, without the lens, to obtain the best definition 

 we are restricted to the use of about two thirds of the grating. 

 In the case of the other grating, we may, without increasing 

 the size of the apparatus, use a lens of 500 centim. focal length; 

 for good definition its aperture should not be more than 



</ 



7 xl25xl0 6 x6 



15 x 10 5 

 Phil. Mag. S. 5. Vol. 15. No. 96. June 1883. 2 H 



