Ex. 3. The logarithmic curve, y = ex. dx ~' dx2 ~' .-. £ — x = —1; .-. * = f + 1; n-y = \ {e-x-e*}, V = g {ex + e-*}, The equation to the catenary; to the lowest point of which the abscissa = — 1, and the ordinate = 1. 325.] The following general properties of caustics by reflexion, formed by a system of parallel rays, deserve consideration. (1) The distance from the incident point in the reflecting curve to the point of intersection of two consecutive reflected rays, is equal to one-fourth of the chord of the circle of curvature at the point of incidence which is parallel to the incident ray. Let Qp, fig. 116, be an incident ray and Pr be the reflected ray, p' being the point where the next consecutive ray cuts it, and which is therefore a point in the caustic; let the circle drawn in the figure, and of which n is the centre and pn is the radius, be the circle of curvature at the point p; then Pf and Pe are the chords of the circle through P which are parallel to the axes of x and y respectively, and let L and K be the bisecting points of Pf and of Pe. Now, according to the notation of Art. 288, Pi is equal to £—x, and Pk to y — ij of that Article; dx2 dy ^ dx* .-. PL = , PK = . (84) d*y dx ••• "' = 2"" Similarly, if the incident rays are parallel to the axis of y, and Q' is the point of the caustic on the reflected ray, it may be shewn by means of equations (82) that , 1 PQ = - PK. The expressions throughout would have assumed a more symmetrical though less simple form if we had not considered J? to be equicrescent. (2) If the radiating point is such that a normal to the reflecting curve can be drawn through it, the caustic corresponding to the point where the normal meets the curve ultimately becomes a semi-cubical parabola. For if the part of the reflecting curve which receives the rays parallel or nearly parallel to the normal through the source of light be taken very small compared to the distance of the origin of light from the curve, the system may be supposed to be one of parallel rays; and also whatever is the reflecting curve we may consider it to be identical with its circle of curvature at the point, so that the problem ultimately becomes, for the small distance, that solved in Ex. 2, Art. 323, wherein we may consider ?j to be small compared with a. .'. 2f = (a* + 2r;#) {aS-r,i}t, ...... = a + ?a*»7*+; and neglecting terms involving powers of Jj higher than those retained, we have g 2£-a = £«M; which is the equation to a semi-cubical parabola; the vertex of which is at a distance „ from the origin. An examination of figs. 117—120 at the point c, renders plain the geometrical form of the problem. 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 = 6, Spo — Rpo = <j>, Poa = 0; then taking o as the origin, and os as the axis of x, the equation to the reflected ray Pr is x sin (0 -f- <£) + y cos (0 + (p) + a sin (f> = 0; or a? (sin 0 cot $ + cos 0) +y (cos0cot4> — sin0) + a = 0; (85) and from the geometry of the figure 6 sin (d — (p) — a sin <p, a + 6 cos 0 2(aa + 3a6cos0 + 262) 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 6 = oo ; 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 which are the equations to an epicycloid; see equations (37), Art. 204, the radii of the fixed and rolling circles being respectively | and |. See fig. 119. (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, y = - (2sin0-sin20) J which are the equations to a cardioid, see Art. 205, the radius of the fixed and generating circles being each ~. o 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 formulae are complicated, we will illustrate the method by the particular case of the logarithmic spiral. In fig. 121 let s 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 B is a point on the caustic j 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 = 6, Rsx = 0; let the equation to the reflecting curve be r = a9; and for convenience of writing, let loge a = A; therefore by Art. 272, Ex. 3, rfr tan Spn = ^ = A, (92) and r = (1 + A*)ip. (93) p' From the geometry, — = sinspz = sin2sPN, 2a = sin(2tan-1A) = = 1 + A 2Ar Also since Srp + Rps + Psr = 180°, .-. sin-1 -4 +2tan-1A + 0'-0 = 180°; differentiating which, p' and 0 varying, but r' and 6' being constant, , > dp -d0 = O; therefore from (92) and (94), 1 + A2 (r'2_p'2)i AT (1+A2)p" .-. r'= (1 + A2)V; which is the equation to a logarithmic spiral, equal to the original one. 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; £, T] the current coordinates of the refracted ray, and therefore of a point on the caustic; p = 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 r' be the distances of the point of incidence from the source of light, and from the point of the caustic; then -r-, ~ are the us as cosines of the angles between the tangent to the curve and the coordinate axes: > those of the angles between the r r incident ray and the coordinate axes; - J*, - those of the angles between the refracted ray and the coordinate axes; (x — a) dx + (y — b) dy ... , „. ., . . 5 r— = the sine of the angle of incidence, rds -—X^ ^X ^—^ — = the sine of the angle of refraction: rds ° therefore by law of refraction, (x-a)dx + (y-b)dy _ (j-x) dx + Q?-y) dy Vdt ~" 7ds ;(a5) which is the equation to the refracted ray, and of which f and rj 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. |