702 Mr. W. Bennett on 
Mr. R. J. Sowter has shown * that for rays on the bounding 
surface of the elliptic-sectioned pencil the eccentric angle (#) 
is constant : this also follows simply from a consideration of 
fig. 1. For if the section by the plane PQR is bounded by 
an ellipse of semiaxes oa and ob, the coordinates of the inter- 
section of the bounding ray AB with this ellipse are o« 
and oy. Since oa will be the radius of the auxiliary cirele, 
cos d = 0” which will have the same value for any point. 
0a’ 
on AB. It is to be noted that ¢ is to be measured always 
from the axis parallel to OX, whether this be the major or 
the minor. 
Prof. 8. P. Thompson has called attention to the rotation 
of the shadow of a straight wire placed across the pencil: 
this also appears from fig. 1. Let a % be the coordinates of 
any point in the plane Z=0, The ray through this point is 
given by 
b—< 
v 
b Op) 

v= 
Reena Sete 
Oe 
a 

te 
| 
Yo 
So that, if 
YY = Me+C 
be the equation of a line in the plane Z=0, the rays passing 
through this line rule the surface 
ay bx 
= Wi +c. 
a—z b—z 


This equation gives also the shadow thrown on a plane 
perpendicular to the axis—that is, the section of the pencil 
by the plane. This is a straight line whose inclination is 
b(a—<) 
a(b—z) 
This inclination varies continually with <. Its tangent 

tation mM. 
=) 0) when 21a; 
= 0 whenz= 8, 
bm 
ae when <= +o. 
The whole amount of rotation between z= and z=—@ 
is 180°, 90° of which occurs between the focal lines. 
Model 3 represents the shadow-surface, and Model 4 shows. 
its relation to the boundary of the pencil. 
* Phil. Mag. Oct. 1903. 
