1903.] On Skew Refraction through a Lens, etc. 517 



x = in equations (4) or (5) of §§ 16, 17, using the values of V,m,' n\ 

 or of /, w, n, there given for 6 = 30°, 60°, . . . 150°. The two 

 which correspond to the points 0° and 180° can in like manner be 

 found by applying equations (3) to values of indefinitely near to 

 and 7T respectively. 



Near 0° we thus obtain, for small 0, 



%-6 = y-l z . 



-0-056626' 0-6505 0-7595 3 



and near 180°, for - - <f> with <f> small, 



x — <f> y + 1 z 



-0-06966c/> 0-7768 0-6297' 

 which, for x = 0, give 



y = 12-50, z = 13-43 ; 7; = -0-66, £ = 18*34.* 



6 = 0°, 



e =180°, 



10-15, 9-04 I 0-785, 13-57. 



The two points thus determined are the ends of the secondary focal 

 line, and the other five points when plotted are found to be sensibty 

 in the straight line joining these two. The tangent of the inclination 

 of the line to the axis of f, as computed from the co-ordinates y f of its 



ends, is * ^ = - -303. 



Sections of the pencil made through the secondary line, are figures 

 of 8 (see 4th row of Plate 9) ; the crossing point of the 8 being 

 the point in which the line meets the section. At the ends of the line, 

 one loop of the 8 vanishes and is replaced by a cusp. The cusps are 

 shown separately with tenfold magnification, in Plate 10. 



23. A simple application of descriptive geometry suffices to show that, 

 in every case of refraction of a homocentric pencil at a spherical 

 surface, all the refracted rays pass through a straight line, namely, 

 the straight line which joins the point-source S to the centre of 

 curvature C of the refracting surface. For, if P be the point of 

 incidence on this surface, PC is the normal, and SPC the plane of 

 incidence. The refracted ray lies in this plane, and therefore meets the 

 line SC. In the case with which we are dealing, S is at infinity, and 

 SC is parallel to the rays within the lens. 



24. Again,- the plane SCP cuts the annulus in a second point P', and 

 the plane of incidence SCP' is identical with SCP. The refracted rays 

 at P and P' lie in this plane, and their intersection is a point in the 

 primary focal line. 



* It is interesting to compare the distance 18 - 34 of the further end of the 

 secondary line Tvith the focal length of the annulus, which is 19"78. 



