54 Mr. A. Whitwell on Refraction 



left to right and the other half from right to left, but it will 

 all pass through the central plane containing the radiant- 

 point and the axis, in the area between these two curves. It 

 is easily seen that the width of the focal area on the axis of 

 a; will be equal to the spherical aberration of the section of 

 the cylinder made by a horizontal plane containing the axis 

 of X, 



We will now consider the focal areas produced in the 

 second symmetrical plane, viz. : the plane containing the 

 radiant-point and normal to the axis of the cylinder. jDraw 

 a similar construction to that shown in fig. 1 for two rays 

 symmetrical with regard to the horizontal plane, and produce 

 the refracted rays backwards till they intersect at the point 

 h in the horizontal plane. This point k will be on a straight 

 line drawn through the radiant-point and parallel to the two 

 normals. 



Consider, first, a thin horizontal slice of the cylinder con- 

 taining the axis of x. For small horizontal aperture the 



TCI 



distance c/= — > By small apertures I mean 



those for which one can neglect the spherical aberration in 

 comparison with the length cf. 



Let 



cf=c. and ol-^cV. 



Then from the figure we have 



a-{-c d^ 



d'=--^ ~ =(tJL — l){a~7-). 



For small horizontal apertures then the focal line is an 

 arc of a circle having its centre at the radiant-point and its 

 radius =(fjL — l) times the distance of the radiant-point from 

 the surface. The focal line is Adrtual for diverging light and 

 real for converging light. If we take two horizontal strips 

 of the cylinder at a distance of h above and below the hori- 

 zontal plane the focal line formed by rays which fall on these 

 two strips will also be a circular arc for small horizontal 

 apertures. Its radius d" is obtained from the relation 



d 



\/a' + h^~ d" 



r \/ rf"^ -L h^ 



