706 Mr. W. Bennett on 
This is the relative retardation of the centre over the edges 
of a piece of the incident wave-front whose width is 2a in a 
direction perpendicular to the plane of fig. 2 and 2a cos @ in 
that plane. 
xe 5) 
Figs 2. 

The changes in the curvatures of the principal sections of 
the wave-front are A and cata That is to say, the lens 
behaves like a bicylindrical lens whose principal converging 
powers are A and A sec? d. This is a standard result derived 
in Heath’s Optics by the use of the characteristic function. 
It only holds, however, for very narrow pencils and thin 
lenses, as it assumes that a is a small quantity of the first, and 
hence # of the second order. 



LXXVIII. Notes on Non-homocentric Pencils, and the Shadows 
produced by them.—II. Shadows produced by Azvially 
Symmetrical Pencils possessing Spherical Aberration. By 
WILLIAM BENNETT™. 
(Plate XXVIL} 
TTENTION has recently been drawn by Prof. 8 P. 
Thompson to some curious shadows produced when a 
straight wire is introduced into an axially symmetrical con- 
verging pencil proceeding from a lens or mirror and possessing 
spher ical aberration. 
The case to be investigated is that of the shadows of a 
straight wire in a pencil produced by the reflexion of 
a par rallel beam at a concave spherical mirror. The wire, and 
* Communicated by the Physical Society : read March 11, 1904, 
