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The objection urged by Mr. Cook against accepting the inductive proof of the actual rate of escape of gases from atmospheres, is analogous to the objection urged by some scientific men when in 1867 I brought forward a proof that in an atmosphere of mixed gases the atmosphere of each gas must have a different limit, the lighter constituents overlapping and extending beyond those which are denser. "Oh!” it was then said, "That can't be the case. It is inconsistent with Dalton's Law of the equal diffusion of gases." Yet I have lived to see my conclusion generally, I believe universally, accepted by physical astronomers; and I look forward with some hope to an ultimate acquiescence in what is now being objected to, in reference to the escape of gases from atmospheres. In both cases the objection rests on the same error-the mistake of hypothesis for theory, and the consequent mistake of a law which is approximate for a law of

nature.

30 Ledbury Road, W., May 12, 1904.

I am, dear Sirs, faithfully yours,
G. JOHNSTONE STONEY.

LXXVII. Notes on Non-homocentric Pencils, and the Shadows produced by them.-I. An Elementary Treatment of the Standard Astigmatic Pencil. By WILLIAM BENNETT †.

A

STANDARD astigmatic pencil is one of which all the rays pass through two focal lines, at right angles to one another and to the axis of the pencil. Let the axis of the pencil be taken as the axis of Z, and let the focal lines be parallel to the axes of X and Y respectively, and at distances a and b from the origin. Then the projections of the pencil on the planes of XZ and YZ are as shown in fig. 1.

It is evident from this figure that the change in the transverse sections of the pencil consists in a uniform stretching or compression in a direction perpendicular to each of the focal lines, the stretchings or compressions proceeding at unequal rates. Thus, if the section is anywhere a conic with axes parallel to the focal lines, it is everywhere so. The pencil is also of rectangular section with sides parallel to the focal lines at all points if at any and this is the section obtained if the extent of the pencil is defined by limiting the lengths of the focal lines.

66

"On the Physical Constitution of the Sun and Stars." By G. Johnstone Stoney, F.R.S. See Proceedings of the Royal Society, No. 105, p. 1 (1898). See especially paragraphs 23, 24, 25.

+ Communicated by the Physical Society: read January 22, 1904.

Any form being assigned to the section by a plane perpendicular to the axis, and the position of the focal lines relative to the plane being defined, it is easy, by drawing

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plans and elevations of the rays through a number of points
on the boundary of the section, to obtain graphically the
section by any other plane parallel to the first. The two
sections being drawn on card or metal, and erected in the
correct relative positions, corresponding points may be
joined by threads or wires and a model of the rays obtained.

Model 1 represents the bounding surface of a pencil of
elliptic section. The construction was simplified by taking
for the first section an ellipse with its horizontal axis twice
the length of its vertical axis, and letting the focal lines.
divide the distance between the two sections into three
equal parts, the horizontal line being nearer the first section;
the second section is an equal and similar ellipse, with its
longer axis vertical.

Model 2 represents a pencil of rectangular section, and was constructed in a similar way.

*

Mr. R. J. Sowter has shown that for rays on the bounding surface of the elliptic-sectioned pencil the eccentric angle (6) 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 intersection of the bounding ray AB with this ellipse are or and oy. Since oa will be the radius of the auxiliary circle, cos o which will have the same value for any point

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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.

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be the equation of a line in the plane Z=0, the rays passing through this line rule the surface

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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

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The whole amount of rotation between = and = -∞ 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.

As this surface is ruled by rays passing through three straight lines, it is a ruled quadric; further, since these three lines lie in parallel planes, it is a hyperbolic paraboloid. One set of generators is the rays. If p, q, r be the direction-cosines of the ray through o, yo, we get from its equations

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Since Yo=ma+c, we have, eliminating 。 and Yo,

aq-mbpcr = 0;

that is, the rays are parallel to the plane

ay—mbx + c = 0.

If c=0, that is if the object-line meets the axis, this becomes

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This case is shown in Model 5, which consists of the b

shadow-surface and the plane y=-mx.

a

The other set of generators are the shadow-lines in planes parallel to

0.

The section by any plane not perpendicular to the axis will be an hyperbola whose equation can be obtained from the equations of the plane and the surface.

If the object-line does not lie in a plane perpendicular to the axis, the ray-surface is an hyperboloid of one sheet, and the shadow on a plane at right angles to the axis becomes an hyperbola. But since a generator of each set passes through every point on the surface, the shadow can always be reduced to a straight line by tilting the plane upon which it is received.

The equation of the ray-surface in this case can be obtained as follows:

The equations to a ray through the point (xo, yo, o) are

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Let the object-line be defined by the equations

x = m + h

y = no + k s

Then, since (o, yo, o) lies on this line, the ray becomes

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Equating the values of obtained from these, we get for the equation of the ruled surface:

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(a−b)xy — (am+1)(:—b) y + (bn + k) (≈—a) x

+(nh—mk) (≈—a)(÷—b) = 0.

This represents a ruled quadric whose section by a plane const. is the rectangular hyperbola whose equation is obtained by substituting for its constant value.

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respectively, each of which passes through one of the focal lines.

The two sets of generators are given by the two pairs of equations:

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respectively.

The first set is the rays, and λ=0. The object-line and the focal lines are members of the second set. The two focal lines are given by μ=0 and μ= respectively, and the mb + h This case is illustrated in model 6.

object-line by μ=

na + k'

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