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 remaining circular, the question can be similarly treated, and it is found that the caustic is an epicycloid in which the 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−1)/4n and a/2n, and in the second, an/(2n + 1) and a/(2n + 1), where a is the radius of the mirror and n the number of reflections.

. 1. . 2. . 3.

The Cartesian equation to the caustic produced by reflection at a circle of rays diverging from any point was obtained by Joseph Louis Lagrange; it may be expressed in the form

{(4c2 − a2)(x2 + y2) − 2a2cx − a2 c2 }3＝27a4c2 y2 (x2 + y2 − 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) cos $)$$2⁄3$ + {(u−p) sin }$2⁄3$＝(2k)$2⁄3$, 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＝∞ the curve reduces to the cardioid or the two cusped epicycloid previously discussed. Other forms are shown in figs. 3, 4, 5, 6. These curves were traced by the Rev. Hammet Holditch (Quart. Jour. Math. vol. i.).

. 4. . 5.

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 primary caustic. It may be shown by geometrical considerations that the secondary caustic is a curve similar to the first 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 limaçon, 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 geometrically that the secondary caustic, if the second medium be less refractive than the first, is an ellipse having the luminous point for a focus, and its centre at the 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 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 locus of the secondary caustic is a Cartesian 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 memoirs by Arthur Cayley.

.—Arthur Cayley’s “Memoirs on Caustics” in the ''Phil. Trans.'' for 1857, vol. 147, and 1867, vol. 157, are especially to be consulted. Reference may also be made to R. S. Heath’s Geometrical Optics and R. A. Herman’s Geometrical Optics (1900).

 CAUTERETS, a watering-place of south-western France in the department of Hautes-Pyrénées, 20 m. S. by W. of Lourdes by rail. Pop. (1906) 1030. It lies in the beautiful valley of the Gave de Cauterets, and is well known for its copious thermal springs. They are chiefly characterized by the presence of sulphur and silicate of soda, and are used in the treatment of diseases of the respiratory organs, rheumatism, skin diseases and many other maladies. Their temperature varies between 75° and 137° F. The springs number twenty-four, and there are nine bathing establishments. Cauterets is a centre for excursions, the Monné (8935 ft.), the Cabaliros (7655 ft.), the Pic de Chabarrou (9550 ft.), the Vignemale (10,820 ft.), and other summits being in its neighbourhood.

 CAUTIN, a province of southern Chile, bounded N. by Arauco, Malleco and Bio-Bio, E. by Argentina, S. by Valdivia, and W. by the Pacific. Its area is officially estimated at 5832 sq. m. Cautin lies within the temperate agricultural and forest region of the south, and produces wheat, cattle, lumber, tan-bark and fruit. The state central railway from Santiago to Puerto Montt crosses the province from north to south, and the Cautin, or Imperial, and Tolten rivers (the latter forming its southern boundary) cross from east to west, both affording excellent transportation facilities. The province once formed part of the territory occupied by the Araucanian Indians, and its present political existence dates from 1887. Its population (1905) was 96,139, of whom a large percentage were European immigrants, principally Germans. The capital is Temuco, on the Rio Cautin; pop. (1895) 7078. The principal towns besides Temuco are Lautaro (3139) and Nueva Imperial (2179), both of historic interest because they were fortified Spanish outposts in the long struggle with the Araucanians.

 CAUTLEY, SIR PROBY THOMAS (1802–1871), English engineer and palaeontologist, was born in Suffolk in 1802. After some years’ service in the Bengal artillery, which he joined in 1819, he was engaged on the reconstruction of the Doab canal, of which, after it was opened, he had charge for twelve years (1831–1843). In 1840 he reported on the proposed Ganges canal, for the irrigation of the country between the rivers Ganges, Hindan and Jumna, which was his most important work. This project was sanctioned in 1841, but the work was not begun till 1843, and even then Cautley found himself hampered in its execution by the opposition of Lord Ellenborough. From 1845 to 1848 he was absent in England owing to ill-health, and on his return to India he was appointed director of canals in the North-Western Provinces. After the Ganges canal was opened in 1854 he went back to England, where he was made K.C.B., and from 1858 to 1868 he occupied a seat on the council of India. He died at Sydenham, near London, on the 25th of January 1871. In 1860 he published a full account of the making of the Ganges canal, and he also contributed numerous memoirs, some written in collaboration with Dr Hugh Falconer, to the Proceedings of the Bengal Asiatic Society and the Geological Society of London on the geology and fossil remains of the Sivalik Hills.

 CAUVERY, or, a river of southern India. Rising in Coorg, high up amid the Western Ghats, in 12° 25′ N. lat. and 75° 34′ E. long., it flows with a general south-eastern direction across the plateau of Mysore, and finally pours itself into the Bay of Bengal through two principal mouths in Tanjore district. Its total length is 472 m., the estimated area of its basin 27,700 sq.m. The course of the river in Coorg is very tortuous. Its bed is generally rocky; its banks are high and covered with luxuriant vegetation. On entering Mysore it passes through a narrow gorge, but presently widens to an average breadth of 300 to 400 yds. Its bed continues rocky, so as to forbid all navigation; but its banks are here bordered with a rich strip of cultivation. In its course through Mysore the channel is interrupted by twelve anicuts or dams for the purpose of irrigation. From the most important of these, known as the Madadkatte, an artificial channel is led to a distance of 72 m., irrigating an area of 10,000 acres, and ultimately bringing a water-supply into the town of Mysore. In Mysore state the Cauvery forms the two islands of Seringapatam and Sivasamudram, which vie in sanctity with the island of Seringam lower down in Trichinopoly district. Around the island of Sivasamudram are the celebrated falls of the Cauvery, unrivalled for romantic beauty. The river here branches into two channels, each of which makes a descent of about 200 m.