
Non-homocentric Pencils. 709 
Let the wire be moved towards the axis until it just 
grazes the passing cuspidal edge. The trace will now 
consist of the branch previously considered and a conjugate 
point which, when the principal focus is passed and the wave- 
front completely unfolded, will be on the other side of the axis. 
If the wire is moved still farther on it is met three times 
by the advancing wave-front: once by the saucer, next by 
the trail immediately behind, and finally by the tail part of 
the trail from the other side. The first two intersections, 
however, are continuous and meet upon the cuspidal edges 
forming a closed curve. The complete shadow now consists 
of an open branch, and a closed branch on the other side of 
the axis. If the wire intersects the axis the shadow passes 
into the @ form particularly noticed by Prof. Thompson. 
The circular part of this is due to a single point on the wave, 
the intersection of the trail with the axis (the normals to the 
wave-surface at this point forming a circular cone), and will 
of course be absent if the trail has not yet met the axis. 
If the wire is placed so near tv the mirror that the cusp 
has not yet begun to form (this stage is not shown in fig. 2), 
the shadow will be single-branched and open on the epposite 
side of the axis. It wiil also be convex towards the axis. 
This, however, becomes closed, and an open branch appears 
on the other side of the axis if the angular aperture of the 
mirror is increased. if, on the other hand, the wire is 
beyond the principal focus, the shadow will be of the form 
first described. 
Fig. 4 (p. 710) isa reproduction of a series of drawings of the 
shadow curves. These were obtained in the following way :— 
Cylindrical coordinates were taken with the axis of the pencil 
for the axis of z and the origin at the centre of curvature, 
the positive direction of z being away from the mirror, and 
¢@ being measured from the plane of the paper. The wire 
lies in the plane z=0 at right angles to the paper and its 
distance from the axis is d. The shadow is considered in the 
Fig. 5. 

plane z=c (fig. 3). The equations of a ray are 
o=const. 
é Sees 
a y= — tan 20 (2+ poe a) 
