Lord Rayleigh's Investigations in Optics. 411 



ration. The distance (A) between the surfaces is given by 



7*=z-z'=G-«)|i (15) 



The equation to the normal to (12) at the point z' } x is 



t-* = t-». ' . 



— 1 x 4/cx* ' 

 so that when f=0, 



. + ^\ 



If the longitudinal aberration be called 8f, 



¥=?-/=ij(l-^) (16) 



Thus by (13) and (14), 



h x 2 



where a denotes the angular semi-aperture. Taking the 

 greatest admissible value of h as equal to \ A, we shall see that 

 hf must not exceed the value given by 



S/=\a-2 (18) 



As a practical example, we may take the case of a single 

 lens of glass collecting parallel rays to a focus. With the 

 most favourable curvatures the longitudinal aberration is about 

 /a 2 ; so that a 4 must not exceed A-r-/. For a lens of 3 feet 

 focus, this condition is satisfied if the aperture do not exceed 

 2 inches. In spectroscopic work the chromatic aberration of 

 single lenses does not come into play, and there is nothing to 

 forbid their employment if the above-mentioned restriction be 

 observed. I have been in the habit of using a plano-convex 

 lens of plate-glass, the curved side being turned towards the 

 parallel light, and have found its performance quite satisfac- 

 tory. The fact that with a given focal length the extreme 

 error of phase varies as the fourth power of the aperture is 

 quite in accordance with practical experience ; for it is well 

 known that the difficulty of making object-glasses for tele- 

 scopes increases very rapidly with the angular aperture. 



When parallel rays fall directly upon a spherical mirror, 

 the longitudinal aberration is only one eighth as great as for 

 the most favourably shaped lens of equal focal length and 

 aperture. Hence a spherical mirror of 3 feet focus might have 

 an aperture of 2^ inches, and the image would not suffer ma- 

 terially from aberration. 



[To be continued.] 



