It will be seen that the object-wire may be moved into the position of any generator of the second set, except the two focal lines, without altering the form of the shadow.

If the plane on which the shadow is taken is not perpendicular to the axis of z, the section is still a conic. If it is turned so as to contain one of the generators of the second set, the section will become two straight lines; the other being the ray which lies wholly in that plane and comes from the point in which the plane is cut by the object-line.

This is an example of a two-part shadow, of which one part is due to a single point of the object.

It is to be noted that although the complete shadow is an hyperbola, it does not follow that both branches will be seen with a pencil limited by any aperture. In fact only one branch will be seen, unless some of the rays which meet it have and some have not passed through one of the focal lines.

The positions of the focal lines produced by reflexion or refraction of a small pencil may be often found by use of the method of sagittæ due to Prof. Thompson. The case of refraction through the centre of a thin lens is taken as an example.

Let r1 and 12 be the radii of curvature of a small thin convex lens of diameter 2a, and refractive index μ. Then the thickness (t) of the lens is given by

Let a small pencil inclined at an angle & to the axis of the lens fall centrally upon it.

Then if a plane wave-front fall upon the glass, the centre of it travels (see fig. 2) through glass for a distance

where sin 1=

sin o
μ

t

cos 41' But this is in a direction making an

angle -1 with the direction of the pencil; so it is only

advanced thereby a distance

t cos $1

cos (-1). Meanwhile

the edges of the wave-front travelling in air have advanced a ; so that the relative retardation of the

με

distance

centre is

cos $1

Phil. Mag. S. 6. Vol. 7. No. 42. June 1904.

3 B

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.

A cos' p

The changes in the curvatures of the principal sections of the wave-front are A and That is to say, the lens behaves like a bicylindrical lens whose principal converging powers are A and A sec2 4. 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 Axially Symmetrical Pencils possessing Spherical Aberration. By WILLIAM BENNETT*.

[Plate XXVII.]

TTENTION has recently been drawn by Prof. S P. A Thompson to some curious shadows produced when a straight wire is introduced into an axially symmetrical converging pencil proceeding from a lens or mirror and possessing spherical aberration.

The case to be investigated is that of the shadows of a straight wire in a pencil produced by the reflexion of a parallel beam at a concave spherical mirror. The wire, and

* Communicated by the Physical Society: read March 11, 1904.

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 from the centre Values of

a

a

2 cos 0

2 cos 0 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 wavefront, which passes through the centre of curvature, to the axis is a (cos + The distance from this wave

1

2 cos

is o).

front to that which passes after reflexion through the principal

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

wave-front first formed. The result is seen in fig. 2, which gives a better view of the form of the waves in this region

Fig. 2

a-20xdist AtoPF

PF

than can be obtained from the smaller figure given by Prof. Wood. The waves are shown as proceeding from a mirror of aperture 52° lying to the left of the diagram; the complete forms are given by Prof. Wood in the paper already referred

Each wave-front is a figure of revolution consisting of a saucer-shaped part in front, bounded by a circular cuspidal edge which is tracing out the caustic surface, and a trailing part behind, which has already passed the caustic. The wave-fronts are given by the equations

where K is a parameter constant over any one wave-front, being, in fact, the optical distance from the incident wavefront through the centre of curvature. These equations, however, are not easily worked with.

Imagine the wire placed so as to meet only the trailing part of the wave-front; as the curvature of the wave-front is not uniform, the trace left upon it by the wire will be distorted as the wave advances, and the shadow will be a single-branched curve on the same side of and farther from the axis. This curve will be concave to the axis in its central parts, and will have two points of inflexion and an asymptote parallel to the wire.

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 wavefront 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 to 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 opposite side of the axis. It will 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) is a 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=b at right angles to the paper and its distance from the axis is d. The shadow is considered in the Fig. 3.

plane

z=c (fig. 3). The equations of a ray are