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CAUSTIC (Gr. rcavvraubs, burning)

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Originally appearing in Volume V05, Page 559 of the 1911 Encyclopedia Britannica.
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CAUSTIC (Gr. rcavvraubs, burning) , that which See also:burns. In See also:surgery, the See also:term is given to substances used to destroy living tissues and so inhibit the See also:action of organic poisons, as in bites, See also:malignant disease and gangrenous processes. Such substances are See also:silver nitrate (lunar caustic), the caustic alkalis (See also:potassium and See also:sodium hydrates), See also:zinc chloride, an See also:acid See also:solution of mercuric nitrate, and pure carbolic acid. In See also:mathematics, the " caustic surfaces " of a given See also:surface are the envelopes of the normals to the surface, or the loci of its centres of See also:principal curvature. In See also:optics, the term caustic is given to the envelope of luminous rays after reflection or See also:refraction; in the first See also:case the envelope is termed a catacaustic, in the second a diacaustic. Catacaustics are to be observed as See also:bright curves when See also:light is allowed to fall upon a polished riband of See also:steel, such as a See also:watch-See also:spring, placed on a table, and by varying the See also:form of the spring and moving the source of light, a variety of patterns may be obtained. The investigation of caustics, being based on the See also:assumption of the rectilinear See also:propagation of light, and the validity of the experimental See also:laws of reflection and refraction, is essentially of a geometrical nature, and as such it attracted the See also:attention of the mathematicians of the 17th and succeeding centuries, more notably See also:John See also:Bernoulli, G. F. de I'Hbpital, E. W. Tschirnhausen and See also:Louis Carre. The simplest case of a caustic See also:curve is when the reflecting surface is a circle, and the luminous rays emanate from a point on the circumference. If in fig.

1 AQP be the reflecting circle caustics having C as centre, P the luminous point, and PQ any by incident See also:

ray, and we join CQ, it follows, by the See also:law of the equality of the angles of incidence and reflection, that the reflection. reflected ray QR is such that the angles RQC and CQP are equal; to determine the caustic, it is necessary to determine the envelope of this See also:line. This may be readily accomplished geometrically or analytically, and it will be found that the envelope is a See also:cardioid (q.v.), i.e. an See also:epicycloid in which the radii of the fixed and See also:rolling circles are equal. When the rays are parallel, the reflecting surface See also:Elie Bocthor (1784—1821) was a See also:French orientalist of Coptic origin. He was the author of a Traite See also:des conjugaisons written in Arabic. and See also:left his See also:Dictionary in MS. remaining circular, the question can be similarly. treated, and it is found that the caustic is an epicycloid in which the See also:radius of the fixed circle is twice that of the rolling circle (fig. 2). The geometrical method is also applicable when it is required to determine the caustic after any number of reflections at a spherical surface of rays, which are either parallel or diverge from a point on the circumference. In both cases the curves are epicycloids; in the first case the radii of the rolling and the fixed circles are a(2n—I)/4n and a/2n, and in the second, an/(2n+I) and a/(2n+I), where a is the radius of the See also:mirror and n the number of reflections. The Cartesian See also:equation to the caustic produced by reflection at a circle of rays diverging from any point was obtained by See also:Joseph Louis See also:Lagrange; it may be expressed in the form +y2 (4c2—a2)(x2~-y2)—2a2cx—a?c2}3 27a4c2y2(x2-c2)2, where a is the radius of the reflecting circle, and c the distance of the luminous point from the centre of the circle. The polar form is {(u+p) See also:cos ZB} I+ {(u—p) See also:sin 1B) a = (2k) a, where p and k are the reciprocals of c and a, and u the reciprocal of the radius vector of any point on the caustic. When c=a or =co the curve reduces to the cardioid or the two cusped epicycloid previously discussed. Other c=;5a c> a forms are shown in See also:figs.

3, 4, 5, 6. These curves were traced by the Rev. Hammet Holditch (Quart. Jour. Math. vol. i.). Secondary caustics are orthotomic curves having the reflected or refracted rays as normals, and consequently the proper caustic curve, being the envelope of the normals, is their evolute. It is usually the case that the secondary caustic is easier to determine than the caustic, and hence, when determined, it affords a ready means for deducing the See also:

primary caustic. It may be shown by geometrical considerations that the secondary caustic is a curve similar to the first See also:positive pedal of' the reflecting curve, of twice the linear dimensions, with respect to the luminous point. For a circle, when the rays emanate from any point, the secondary caustic is a limacon, and hence the primary caustic is the evolute of this curve. The simplest instance of a caustic by refraction (or diacaustic) is when luminous rays issuing from a point are refracted at a straight line. It may be shown geometric-ally that the secondary caustics caustic, if the second by See also:mica,. See also:medium be less refrac- See also:don. tive than the first, is an See also:ellipse having the luminous point for a See also:focus, and its centre at the See also:foot of the perpendicular from the luminous point to the refracting line. The evolute of this ellipse is the caustic required.

If the second medium be more highly refractive than the first, the secondary caustic is a See also:

hyperbola having the same focus and centre as before, and the caustic is the evolute of this curve. When the refracting curve is a circle and the rays emanate from any point, the See also:locus of the secondary caustic is a Cartesian See also:oval, and the evolute of this curve is the required diacaustic. These curves appear to have been first discussed by Gergonne. For the caustic by refraction of parallel rays at a circle reference should be made to the See also:memoirs by See also:Arthur See also:Cayley.

End of Article: CAUSTIC (Gr. rcavvraubs, burning)

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