114 



Miss A. Everett on a Projective Theorem 



have only to replace the radins o£ the sphere by the proper 



radius of curvature, find the 



tions, and then the required 



ray will be the ray of which these paths are the projections 



The refracted ray is usually found in terms of its inclination 



rerracted paths of the projec- 

 refracted path of the oblique 



Fie. 1. 



to the optic axis, and the " schnittweite " or distance from 

 the last surface of the intersection of ray and axis. If these 

 quantities be S, s for the sagittal projection, T, t for the 

 tangential, and the planes of projection be taken as the xy 

 and xz planes, then the required ray is given as the line of 

 intersection of the planes 



x — s — y cotS, x — t = z cotT, 



or as 



y + s tan S z + 1 tan T 



tan 8 



tan T 



In general the theorem cannot be applied to tracing a ray 

 through a series of surfaces, because the principal sections 

 will vary from surface to surface. Culmann, however, 

 applies it to a series of surfaces in two special cases : — 

 (I.) A series of non-spherical surfaces which are all normal 

 to the chief ray, and have their principal sections coincident. 

 (II.) Two infinitely thin non-spherical systems with principal 

 sections not coincident. As an instance of (I.), Culmann 

 mentions crossed cylinders. As another instance may be 

 mentioned a coaxial series of tores having the centres of 

 their generating circles all lying on a right circular cone 

 with its axis on the axis of revolution (a tore being defined 

 as the surface generated by a circle revolving about an axis 

 in its plane). Thus in an axial section (fig. 2) the centres 

 of the generating circles would lie on a pair of straight lines 

 meeting on the axis, each line being a common normal to all 



