Page:The New International Encyclopædia 1st ed. v. 05.djvu/787

CTIRTIUS. 679 CURVE. Bonn, but on the death of Victor ^[cycr, in the same year, accepted the professorsliip of chemis- try at Heidelberg. Curtius discovered new series of important compounds of nitrojjfn, including hydrazine and many of its derivatives, and the diazo-derivatives of the fatty series of organic compounds. CU'KXJLE CHAIR (translation of Lat. sella ciiiulis, from nello, seat, from scdcre, to sit, and cunilis, curule, from ciirrus, chariot, from cur- tcre, to run). The chair of state, equivalent to a throne, among the early Romans. JvTo one ex- <^^ept consuls, i)r;rtors. and a few others high in authority was permitted to occupy it. The chair was usually ornamented with gold and other precious work. CURVATURE. See Curve. CURVATURE OF FIELD OF A LENS. See Light for general discussion of the jjroper- tics of lenses. CURVATURE OF THE EARTH'S SUR- FACE. See Curve. CURVE (OF. coiirhe, corhc, Fr. co»r5e, Sp., Port., It. curvo, from Lat. cifrcHs, curved, OChureh Slav, krivii, bent, Lith. kreiras, ci'ooked). In common language, a line that constantly de- parts from a fixed direction. In analytic geome- try, however, the word curve is connnonlj' used to designate the locus of a point moving accord- ing to any definite law, and hence to include the straight line. If the statement of the law according to which the point moves can be translated into an equation or equations be- tween the coordinates (q.v. ) of the moving point, these equations may be used to represent the curve — e.g. the circle is the locus of a point moving in a plane at a constant finite distance from a fixed point in that plane, and its equa- tion is X- + y' ^ r'. ( See Coordisateis. ) If the curve possesses the property of continuity (q.v.) it is precisely definaVde at every point, although it may contain singularities. The form of a eun'e corresponds to the nature of its equation; hence a curve may l)e designated as algebraic or transcendental according as its equation con- sists of algebraic or transcendental functions of the coordinates; for example, the conic sections are algebraic curves, and the cycloid, the loga- rithmic spiral, and the catenary are transcenden- tal curves. Algeliraic curves are fundamentally grouped into orders and classes, according to Newton's classification. The order of a plane curve is determined by the number of points, real or imaginary, in which it intersects an}' line in its plane. Curves which cut such lines in two points are called curves of the second order; those which cut the lines in three points curves of the third order, and so on — e.g. the conic sections are all curves of the second order, and cubic curves are of the third order. The straight line is the only line of the first order. Similarly the order of an algebraic curve in space depends upon the number of points in which it cuts any plane. The class of an alge- braic plane curve is determined by the number of tangents, real or imaginary, which can be drawn to it from any point in its plane. If two tangents are possible it is a curve of the second class, if three are possible, a curve of the third class, and so on — e.g. the conic sections are curves of the second class; the cissoid (q.v.) is of the third class. Similarly the class of a space curve is given by the numl)cr of tangent planes which can be drawn containing any fixed line. The class of a plane curvo dcjicnds directly upon its order when no singularities exist. If n is the order and c the class, c:=n(ji — 1). Thus a conic with no singular points is of the second class, since c = 2(2— 1) =2; the cubic is of the sixth class, since c = .'J(.3 — 1) =: (i. But singularities tend to diminish the class. Pliickcr gave six eipialioiis cnjuiccting the order, class, nun.ber of d()ul)le jioints, numl)cr of double t.ingents, numlier of stationary [wints, and num- ber of stationary tangents from which, if any three of those numbers are given, the otluT tliree may be obtained. The one directly connecting the order and cla.ss is c = n' — n — 2d — Sp, in which c is the class, n the order, d the number of double [jointsj and p the number of stationary points. Thus, a cubic with one double point is a curve of the fourth cla.ss, since c = 9 — 3 — 2 = 4. By the aid of covariants (sec Forms), the class of a curve can be determined directly. Singularities. ( 1 ) An algebraic curve whose equation is y=r-f(x) is convex or concave downward, according as —r4 is positive or negative. » (2) A point of inflection is one at which the tangent to the curve takes a limiting position — that is, the point of contact of a stationary tangent, at which -—^ = or ao . See Curve OF Sines. (3) A multiple point is one at which more than one tangent exists — that is, a point for which ~f- has more than one value. Two values dx determine a double point, three values a triple point, and so on. A multiple point is also called a node or crunode. Multiple points of the third order are divided into classes according to the relative number of cusps and crunodcs involved. (4) When two branches of a curve have a common tangent at a point, but do not pass through the point, they are said to form a cusp, called also a spinode or stationary point. See Cissoid; Conchoid. (5) An Isolated point whose coordinates satisfy the equation of the curve is called a conjugate point or acnode. An acnode is a multiple point at which the tangents are imaginary. A node or conjugate point corresponds to a double tan- gent, and a cusp to a stationary tangent. Curvature. The curature of a jjlane curve at any point is its tendency to depart from a tangent to the curve at that point. In the circle this deviation is constant, as the curve is per- fectly symmetrical round its centre. The curva- ture of a circle varies, however, inversely as the radius — that is, it diminishes at the .same rate as the radius increases. The reciprocal of the radius is therefore taken as the measure of the curvature of a circle. A straight line may be considered a circle of infinite radius and as hav- ing no cui'vature, since — = 0. The constancy 00 of curvature in the circle suggests an absolute measure of curvature at any point in any other curve, for whatever bo the curvature at that point a circle can be found of the same cun'a- ture. The radius of this circle is called the radius of curvature for that point; and the