Non-homocentric Pencils. 707 
the plane on which the shadow is received, will be taken 
perpendicular to the axis of the pencil. Let a be the radius 
of curvature of the mirror (fig. 1), then a ray which meets 

the mirror at an angular distance @ from its pole will, after 
reflexion, make an angle 20 with the axis, and will meet it 
at a distance Remmestes from the centre Values of =n 
2 cos 0 2 cos 6 
being tabulated, it is easy to make a drawing of a part of the 
pencil to any scale. The most important shadows are those 
produced when the wire is near the principal focus and ° 
between it and the mirror ; and in the consideration of these 
it is convenient to think of the form of the wave-front which 
meets the wire, and of the trace left upon it after it has 
passed the wire. A method for drawing the wave-front 
has been given by Prof. R. W. Wood*; but as this method 
is inconvenient for large-scale drawings, a modification of it 
was adopted. The optical distance from the incident wave- 
front, which passes through the centre of curvature, to the 
: ) The distance from this wave- 
2 cos @ 
front to that which passes after reflexion through the principal 

axis 1s a (cos pee 
focus is 2 . A reflected ray therefore meets the latter wave- 
front at a distance a(5 — cos 6-5-5) beyond its inter- 
2 2cos0/  : 
section with the axis. The values of this quantity were 
calculated for different values of 20, and measured off on a 
large scale drawing of the rays near the principal focus. 
The other wave-fronts were obtained by measuring back a 
series of equal distances along the rays, starting from the 
* Phil. Mae. [5] vol. 1. (July 1900). 
3B2 
