Page images
PDF
EPUB

326.] The general form of the equation to a caustic of a circle by reflexion may be most conveniently determined as follows:

In fig. 118 let s be the source of light, and sp the incident type-ray, and PR the reflected type-ray, o being the centre of the circle. Let oa = a, os = b, SPO = RPO = 4, POA = 0; then taking o as the origin, and os as the axis of x, the equation to the reflected ray PR is

or

x sin (0+)+ y cos (0+ $) + a sin

= 0;

x (sin cot + cos 0) + y (cos e cot — sin 0) + a = 0; (85) and from the geometry of the figure

[blocks in formation]

x (a sine+b sin 20) + y (a cos 0 + b cos 20) + ab sin 0 = 0. (87)

Differentiating with respect to 0, we have

x (a cos0+2b cos 20)—y (a sin 0+2b sin 20) + ab cos 0 = 0; (88) whence, eliminating from (87) and (88), we have

[blocks in formation]

which are the equations to the caustic in terms of a subsidiary angle 0. In two cases they reduce themselves to the equations of an epicycloid.

(1) Let b = ∞; so that the source of light is at an infinite distance, and we have a system of parallel rays incident parallel to the axis of x. Then

[blocks in formation]

which are the equations to an epicycloid; see equations (37), Art. 204, the radii of the fixed and rolling circles being respect

[blocks in formation]

(2) Let b = a; in which case the source of light is at the extremity of the diameter of the circle, see fig. 120, and the equations (69) become,

[blocks in formation]

which are the equations to a cardioid, see Art. 205, the radius

of the fixed and generating circles being each

a

3

327.] Caustics by reflexion from curves expressed in terms of polar coordinates, and which have the origin of light at the pole, may be determined in the following manner; but as the general formulæ are complicated, we will illustrate the method by the particular case of the logarithmic spiral.

In fig. 121 lets be the pole of the spiral and the source of light, sp the incident, PR the reflected ray. Let R be the point in which two successive rays intersect, wherefore R is a point on the caustic; and it is also to be observed that PR is a tangent to the caustic. Let SP r, SY = p; SR = r', sz = p′; psx = 0, RSX = 0; let the equation to the reflecting curve be r = ao; and for convenience of writing, let log, a = A; therefore by Art. 272, Ex. 3,

[blocks in formation]

dr rdo

= A,

r = (1 +▲2)*p.

= sin SPZ = sin 2 SPN,

=sin (2 tan-1A) =

2Ar
1 + A2 *

(92)

(93)

[blocks in formation]
[blocks in formation]

SRP RPS PSR =

180°,

[blocks in formation]

differentiating which, p' and 0 varying, but r' and 'being

constant,

dp'
(r'2 — p ́2) à

— d0 = 0;

therefore from (92) and (94),

[merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small]

which is the equation to a logarithmic spiral, equal to the original one.

η

328.] We proceed now to consider some of the more general properties of caustics by refraction. Let (a, b) be the source of light; (x, y) the point on the surface at which the ray is incident; έ, the current coordinates of the refracted ray, and therefore of a point on the caustic; μ the refractive index, that is, the ratio of the sine of the angle of incidence to that of the angle of refraction. Let y = f(x) be the equation to the section by the paper of the bounding surface of the refractive medium, the surface being perpendicular to the paper; let r and be the distances of the point of incidence from the source of light, and from the point of the caustic; then cosines of the angles between the tangent to the curve and the

x-a y -b

dx dy

ds' ds

are the

those of the angles between the n-y those of

coordinate axes:
incident ray and the coordinate axes;

the angles between the refracted ray and the coordinate axes;

[blocks in formation]

which is the equation to the refracted ray, and of which έ and ૐ η are the current coordinates; the envelope therefore may be found by eliminating x and y between the equation to the refracting curve, the equation (95), and its differential formed by making x and y to vary.

[ocr errors]
[merged small][merged small][merged small][ocr errors][ocr errors][merged small][merged small][merged small][merged small][merged small]

therefore r+ur' is a constant, or a maximum, or a minimum; but it cannot be a maximum, for such a value would be inconsistent with the geometrical possibility of the problem: therefore it is in general a minimum, and may sometimes be constant; the former case is that of an ordinary caustic; in the latter the refracted rays converge to a single focus.

n

329.] Hence also we may prove that all caustics are rectifiable. Let έ, ʼn be the current coordinates, and do the length of the element of the curve of the caustic, so that do2 = dŋ2 +d§2 ; dę dn therefore are the cosines of the angles made by its do' do

tangent with the coordinate axes;

[merged small][merged small][merged small][ocr errors][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small]

r'dr' = (§ — x) (d§ — dx) + (n − y) (dn— dy) ;

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small]

(98)

(99)

An expression exactly analogous to that of Art. 292, and to which therefore a similar mode of explanation is applicable; and therefore the length of the caustic curve is equal to that of two straight lines increased by a constant which is to be determined by the data of the particular problem; but in all cases, if 01 r1rí', O2 r2 rá', represent two sets of corresponding values,

μ (02-01) = r 2 − r1 + μ (r2' -rí).

The law of refraction becomes that of reflexion, if μ= -1; and therefore the properties of caustics by refraction proved above are likewise true of caustics by reflexion; attention must however be paid to an ambiguity of sign, of which no notice has been taken in the preceding investigation.

330.] To determine the caustic by refraction of rays refracted at a plane surface; see fig. 122.

Let s be the source of light; SP the incident ray; RPr the refracted ray; As = a, AP = y; therefore the equation to the refracted ray is

n-y-tan PTA.έ,

r being the point where PR intersects Asc.

And since sin SPN = μ

sin ren,

... sin PSA μ sin PTA,

[blocks in formation]

Differentiating which with respect to y, and reducing

μ2 a2 + (μ2 − 1) y2 = §* μŝaŝ,

[merged small][merged small][subsumed][ocr errors][subsumed][ocr errors]

(100)

(101)

[merged small][ocr errors][ocr errors][subsumed][merged small]

which is the equation to the evolute of a hyperbola or of an ellipse according as μ is greater or less than unity.

PRICE, VOL. I.

38

« PreviousContinue »