# Curves Sentence Examples

curves
• She had soft curves in all the right places.

• The curves corresponding to the above expressions are plotted in fig.

• The third section of 113 m., from Paks to the mouth of the Drave, differed from the others and made innumerable twists and curves.

• The curves on railways are either simple, when they consist of a portion of the circumference of a single circle, or compound, when they are made up of portions of the circumference of two or more circles of different radius.

• On some of the earlierEnglish main lines no curves were constructed of a less radius than a mile (80 chains), except at places where the speed was likely to be low, but in later practice the radius is sometimes reduced to 40 or 30 chains, even on high-speed passenger lines.

• The resistance to motion round a curve has not been so systematically studied that any definite rule can be formulated applicable to all classes of rolling stock and all radii of curves.

• A general result could not be obtained, even from a large number of experiments, because the resistance round curves depends upon so many variable factors.

• Rate at which work is done against the resistances given by the curves fig.

• As given by the Barbier curves in fig.

• Using the curves of fig.

• Thus the length of the body was limited, for to increase it involved an increase in the length of the rigid wheel base, which was incompatible with smooth and safe running on curves.

• In the great continental basin there are long lines with easy gradients and curves, while in the Allegheny and Rocky Mountains the gradients are stiff, and the curves numerous and of short radius.

• On the lines actually authorized by the Board of Trade under the 1896 act the normal minimum radius of the curves has been fixed at about 600 ft.; when a still smaller radius has been necessary, the speed has been reduced to 10 m.

• A straight length of not less than 60 metres for the largest gauge and 40 metres for the smallest must be made between two curves having opposite directions.

• The chief difference between the first three types lies in the weight of rails and rolling stock and in the radius of the curves.

• To these curves, which were also applied to effect some quadratures, Evangelista Torricelli gave the name of "Robervallian lines."

• Such curves are given by the equation x 2 - y 2 = ax 4 -1bx 2 y 2 +cy 4 .

• These curves are instances of unicursal bicircular quartics.

• In it Maclaurin developed several theorems due to Newton, and introduced the method of generating conics which bears his name, and showed that many curves of the third and fourth degrees can be described by the intersection of two movable angles.

• He also gave in his Fluxions, for the first time, the correct theory for distinguishing between maxima and minima in general, and pointed out the importance of the distinction in the theory of the multiple points of curves.

• The shore line curves away, beyond these, westward to the Start and eastward to Portland - both visible from Sidmouth beach.

• The whole plan is drawn from three centres, the outer portion of the curves being arcs of a larger circle than the one used for the central portion; the complete circle of the orchestra is marked by a sill of white limestone, and greatly enhances the effect of the whole.

• A volume entitled Opera posthuma (Leiden, 1703) contained his "Dioptrica," in which the ratio between the respective focal lengths of object-glass and eye-glass is given as the measure of magnifying power, together with the shorter essays De vitris figurandis, De corona et parheliis, &c. An early tract De ratiociniis tin ludo aleae, printed in 16J7 with Schooten's Exercitationes mathematicae, is notable as one of the first formal treatises on the theory of probabilities; nor should his investigations of the properties of the cissoid, logarithmic and catenary curves be left unnoticed.

• The general character of curves of magnetization and of induction will be discussed later.

• A notable feature in both classes of curves is that, owing to hysteresis, the ascending and descending limbs do not coincide, but follow very different courses.

• After a few repetitions of the reversal, the process becomes strictly cyclic, the upward and downward curves always following with precision the paths indicated in the figure.

• Steinmetz's formula may be tested by taking a series of hysteresis curves between different limits of B,' measuring their areas by a pianimeter, and plotting the logarithms of these divided by 47r as ordinates against logarithms of the corresponding maximum values of B as abscissae.

• Curves of magnetization (which express the relation of I to H) have a close resemblance to those of induction; and, indeed, since B = H+47r1, and 47rI (except in extreme fields) greatly exceeds H in numerical value, we may generally, without serious error, put I = B /47r, and transform curves of induction into curves of magnetization by merely altering the scale to which the ordinates are referred.

• A study of such curves as these reveals the fact that there are three distinct stages in the process of magnetization.

• But at any intermediate frequency the ascending and descending curves of magnetization will enclose a space, and energy will be dissipated.

• Honda, measured the changes of length of various metals shaped in the form of ovoids instead of cylindrical rods, and determined the magnetization curves for the same specimens; a higher degree of accuracy was thus attained, and satisfactory data were provided for testing theories.

• Austin, who found continuous elongation with increasing fields, the curves obtained bearing some resemblance to curves of magnetization.

• Specimens of curves showing the relation of induction to magnetic field at various temperatures, and of permeability to temperature with fields of different intensities, are given in figs.

• The paper contains tables and curves showing details of the magnetic changes, sometimes very complex, at different temperatures and with different fields.

• After this operation had been repeated a few times the iron was found to have acquired a stable condition, and the curves corresponding to the two temperatures became perfectly definite.

• Most of the permeability-temperature curves were more or less convex towards the axis of temperature, and in all the experiments except those with annealed iron and steel wire, the permeability was greatest at the lowest temperature.

• The curves given by Houllevigue for the relation of thermo-electric force to magnetic field are of the same general form as those showing the relation of change of length to field.

• Rhoads obtained a cyclic curve for iron which indicated thermo-electric hysteresis of the kind exhibited by Nagaoka's curves for magnetic strain.

• Further, it was shown that the thermo-electric curves were modified both by tensile stress and by annealing in the same manner as were the change-of-length curves, the modification being sometimes of a complex nature.

• The essential point in his advance on Euler's mode of investigating curves of maximum or minimum consisted in his purely analytical conception of the subject.

• The Umzimkulu river rises in Bamboo Castle, in the Drakensberg, and, with bolder curves than the Umkomaas, runs in a course generally parallel with that stream S.E.

• As might be expected in a country possessing the physical features of Natal, the gradients and curves are exceptionally severe.

• The Kobdo river, which rises in the Dain-gol (7060 ft.) in the Ektagh Altai, winds in great curves across the plateau, and enters Lake Kara-usu (3840 ft.), which also receives the Buyantu, an outflow from Lake Kobdo, and is connected by a small river with another large lake, Durga-nor, situated a score of miles to the east.

• South of this point the coast curves outwards and is broken by peninsulas and indentations; to the north it is concave and bordered in many places by dunes and lagoons.

• The coast, which curves to the N.E., is marked by a line of sandhills covered with thick bush and rising in places to a height of 500 ft.

• Sharp curves should be avoided, especially for mechanical haulage.

• The ropes are supported between the rails and guided on curves by rollers and sheaves.

• In the endless-rope systems cars run singly or in short trains, curves are disadvantageous, unless of long radius, speed is relatively slow, and branch roads not so easily operated as with tail-rope.

• These curves attracted much attention and were discussed by John Bernoulli, Leibnitz, Huygens, David Gregory and others.

• The mechanical properties of the curves are treated in the article MECHANIcs,where various forms are illustrated.

• The curves 0 = constant and 4, = constant form an orthogonal system; and the interchange of 0 and 4, will give a new state of uniplanar motion, in which the velocity at every point is turned through a right angle without alteration of magnitude.

• The shore line of the bay is broken by large, deeply indented bays (that of Jurujuba being nearly surrounded by wooded hills), shallow curves and sharp promontories.

• The curves Pa4Q, having a minimum at a4, Pa3Q, having a maximum at a 31 and Pa 5 Q, with neither a maximum nor minimum, correspond to the types i., ii., iii.

• The northern part of Coatue Beach is known as Coskata Beach, and curves to the N.W.; near its tip is Great Point, where a lighthouse was first built in 1784.

• The shapes of the body and lid corresponded so intimately that, whereas the lid could be slipped on easily and smoothly without any attempt to adjust its curves to those of the body, it always fitted so closely that the box could be lifted by grasping the lid only.

• Massive, towering roofs, which impart an air of stateliness even to a wooden building and yet, by their graceful curves, avoid any suggestion of ponderosity, were still confined to Buddhist edifices.

• The site of the town curves round the harbour, between it and the strongly fortified hills of Antennamare, the highest point of which is 3707 ft.

• The two sloping lines cutting at the eutectic point are the freezing-point curves of alloys that, when they begin to solidify, deposit crystals of lead and tin respectively.

• But the curves are not always so simple as the above.

• Many other inter-metallic compounds have been indicated by summits in freezing-point curves.

• Of these, the first three are well indicated on the freezing-point curves.

• The intermediate summits occurring in the freezing-point curves of alloys are usually rounded; this feature is believed to be due to the partial decomposition of the compound which takes place when it melts.

• If now we cut the freezing-point surface by planes parallel to the base ABC we get curves giving us all the alloys whose freezing-point is the same; these isothermals can be projected on to the plane of the triangle and are seen as dotted lines in fig.

• The importance is now widely recognized of considering the mechanical properties of alloys in connexion with the freezing-point curves to which reference has already been made, but the subject is a very complicated one, and all that need be said here, is that when considered in relation to their meltingpoints the pure metals are consistently weaker than alloys.

• It mounts first westwards to an open space, then turns eastwards till it reaches the eastern end of the terrace wall that supports the temple, and then turns again and curves up north and then west towards the temple.

• In his investigations respecting cycloidal lines and various spiral curves, his attention was directed to the loxodromic and logarithmic spirals, in the last of which he took particular interest from its remarkable property of reproducing itself under a variety of conditions.

• To the north-west, beyond the Tal-i-Bangi, the magnificent outlines of the Mosalla filled a wide space with the glorious curves of dome and gateway and the stately grace of tapering minars, but the impressive beauty of this, by far the finest architectural structure in all Afghanistan, could not be permitted to weigh against the fact that the position occupied by this pile of solid buildings was fatal to the interests of effective defence.

• This method, which is the oldest, is best adapted for ways that are nearly level, or when many branches are intended to be worked from one engine, and can be carried round curves of small radius without deranging the trains; but as it is intermittent in action, considerable engine-power is required in order to get up the required speed, which is from 8 to ro m.

• This system presents the greatest advantages in point of economy of driving power, especially where the gradients are variable, but is expensive in first cost, and is not well suited for curves, and branch roads cannot be worked continuously, as a fresh set of pulleys worked by bevel gearing is required for each branch.

• A third group, of increasing importance, comprises cases in which curves or surfaces arise out of the application of graphic methods in engineering, physics and statistics.

• In elementary geometry we deal with lines and curves, while in mensuration we deal with areas bounded by these lines or curves.

• Thus Nicomedes invented the conchoid; Diodes the cissoid; Dinostratus studied the quadratrix invented by Hippias; all these curves furnished solutions, as is also the case with the trisectrix, a special form of Pascal's limacon.

• Fourier's theorem asserts that such a curve may be built up by the superposition, or addition of ordinates, of a series of sine curves of wave-lengths X, IX, 3A, 4A...

• The plate is then bowed at the edge and is thrown into vibration between nodal lines or curves and the sand is thrown from the moving parts or ventral segments into these lines, forming " Chladni's figures."

• The three lightly W2 dotted parabolas are the curves of maximum moment for each of the loads taken separately.

• The three heavily dotted curves are curves of maximum moment under each of the loads, for the three loads passing over the bridge, at the given distances, from left to right.

• The single street of the island curves from the gateway up to the abbey, ending in flights of steps leading to the donjon or chatelet.

• The accuracy of a meter is tested by drawing calibration curves showing the percentage departure from absolute accuracy in its reading for various decimal fractions of full load.

• He extended the "law of continuity" as stated by Johannes Kepler; regarded the denominators of fractions as powers with negative exponents; and deduced from the quadrature of the parabola y=xm, where m is a positive integer, the area of the curves when m is negative or fractional.

• The appreciation of light and well-proportioned curves and the skilful employment of well-contrived pierced work are conspicuous features.

• Hence we see that if one unit is derived from another it may be possible, by the similarity or difference of the forms of the curves, to discern whether it was derived by general consent and recognition from a standard in the same condition of distribution as that in which we know it, or whether it was derived from it in earlier times before it became so varied, or by some one action forming it from an individual example of the other standard without any variation being transmitted.

• The requirements of an elongate body moving through the resistant medium of water are met by the evolution of similar entrant and exit curves, and the bodies of most swiftly moving aquatic animals evolve into forms resembling the hulls of modern sailing yachts (Bashford Dean).

• The cuesta begins where its determining limcstone begins, in west-central New York; there it separates the lowlands that contain the basins of lakes Ontario and Erie; thence it curves to the north-west through the province of Ontario to the belt of islands that divide1 Georgian Bay from Lake Huron; then westward throtigh the land-arm between lakes Superior and Michigan, and south-westward into the narrow points that divide Green Bay from Lake Michigan, and at last westward to fade away again with the thinning out of the limestone; it is hardly traceable across the Mississippi river.

• Obviously these equations show that the curves intersect in four points, two of which lie on the intersection of the line, 2 (g - g')x +2 (f - f')y+c - c'=o, the radical axis, with the circles, and the other two where the lines x2+y2= (x+iy) (x - iy) =o (where i = - - I) intersect the circles.

• Essentially, therefore, Descartes's process is that known later as the process of isoperimeters, and often attributed wholly to Schwab.2 In 16J5 appeared the Arithmetica Infinitorum of John Wallis, where numerous problems of quadrature are dealt with, the curves being now represented in Cartesian co-ordinates, and algebra playing an important part.

• It is clearly the form of the fundamental property (expressed in the terminology of the "application of areas") which led him to call the curves for the first time by the names parabola, ellipse, hyperbola.

• The kaleidophone, intended to present visibly the movements of a sonorous body, consisted of a vibrating wire or rod carrying a silvered bead reflecting a point of light, the motions of which, by persistence of the successive images on the retina, were thus represented in curves of light.

• A spectroscope may be compared to a mechanical harmonic analyser which when fed with an irregular function of one variable represented by a curve supplies us with the sine curves into which the original function may be resolved.

• The angle between a line and a curve (mixed angle) or between two curves (curvilinear angle) is measured by the angle between the line and the tangent at the point of intersection, or between the tangents to both curves at their common point.

• The slope of these curves is determined by the so-called "latent heat equation" FIG.

• For the quantitative study of such systems in detail it is convenient to draw plane diagrams which are theoretically projections of the curves of the solid phase rule diagram on one or other of these planes.

• If the two substances are soluble in each other in all proportions at all temperatures above their melting points we get a diagram reduced to the two fusion curves cutting each other at a nonvariant point.

• Thus in interpreting complicated freezing point curves, we must look for chemical compounds where the curve shows a maximum, and for a eutectic or cryohydrate where two curves meet at a minimum point.

• With the most likely forms for the energy curves we get the accompanying diagrams for the relation between freezing point and concentration.

• It will be noticed that in all these theoretical curves the points of initial fusion and solidification do not in general coincide; we reach a different curve first according as we approach the diagram from below, where all is solid, or from above, where all is liquid.

• The freezing and melting point curves are exactly similar to theoretical curves of fig.

• When a crystal of the solid phase is present the equilibrium of a solution is given by the solubility curves we have studied.

• When two substances are soluble in each other in all proportions, we get solubility curves like those of copper and silver shown in fig.

• We should expect to find supersolubility curves lying below the solubility curves, and this result has been realized experimentally for the supersolubility curves of mixtures of salol (phenyl salicylate) and betol (/3-naphthol salicylate) represented by the dotted lines of fig.

• The slope of the temperature vapour pressure curves in the neighbourhood of the freezing point of the solvent is given by the latest heat equation.

• Callendar finds that five molecules of water in the case of cane-sugar or two molecules in the case of dextrose are required to bring the curves into conformity with the observations of Berkeley and Hartley, which in fig.

• The axes will take up any position, and consequently give rise to a continuous series of parhelia which touch externally the inner halo, both above and below, and under certain conditions (such as the requisite altitude of the sun) form two closed elliptical curves; generally, however, only the upper and lower portions are seen.

• Thence it curves southwest, past Potsdam and Brandenburg, traversing another chain of lakes, and finally continues north-west until it joins the Elbe from the right some miles above Wittenberge after a total course of 221 m.

• The plumage is white, except the primaries, which are black, and a black plume, formed by the secondaries, tertials and lower scapulars, and richly glossed with bronze, blue and green, which curves gracefully over the hind-quarters.

• Monge's memoir just referred to gives the ordinary differential equation of the curves of curvature, and establishes the general theory in a very satisfactory manner; but the application to the interesting particular case of the ellipsoid was first made by him in a later paper in 1795.

• These freezing-point curves and transformation curves thus divide the diagram into 8 distinct regions, each with its own specific state or constitution of the metal, the molten state for region 1, a mixture of molten metal and of solid austenite for region 2, austenite alone for region 4 and so on.

• One of the best methods of studying the flow of heat in this case is to draw a series of curves showing the variations of temperature with depth in the soil for a series of consecutive days.

• The curves given in fig.

• The mean temperature gradient is found by plotting the curves for each day from the daily observations.

• We thus obtain the simple equation k'(de'/dx') - k"(de"/dx") =c (area between curves)/(T - T'), (4) by means of which the average value of the diffusivity klc can be found for any convenient interval of time, at different seasons of the year, in different states of the soil.

• The dotted boundary curves have the equation 0 =omx, and show the rate of diminution of the amplitude of the temperature oscillation with depth in the metal.

• Among other modern thoroughfares, the Via di Circonvallazione a Monte, laid out since 1876 on the hills at the back of the town, leads by many curves from the Piazza Manin along the hill-tops westward, and finally descends into the Piazza Acquaverde; its entire length is traversed by an electric tramway, and it commands magnificent views of the town.

• Fritz (2) has, however, drawn a series of curves which are believed to give a good general idea of the relative frequency of aurora throughout the northern hemisphere.

• Fritz' curves, shown in the illustration, are termed isochasms, from the Greek word employed by Aristotle to denote aurora.

• The XIXth Dynasty, at its best under Seti I., could only excel in high finish of smoothness and graceful curves; life, character, meaning, had vanished.

• In the geometry of plane curves, the term parabola is often used to denote the curves given by the general equation a' n x n = ym+n, thus ax= y 2 is the quadratic or Apollonian parabola; a 2 x = y 3 is the cubic parabola, a 3 x = y4 is the biquadratic parabola; semi parabolas have the general equation ax n-1 = yn, thus ax e = y 3 is the semicubical parabola and ax 3 = y 4 the semibiquadratic parabola.

• These curves were investigated by Rene Descartes, Sir Isaac Newton, Colin Maclaurin and others.

• Diverging parabolas are cubic curves given by the equation y 2 = 3 -f-bx 2 -cx+d.

• Though something in the grotesque dragons of the base recalls the Byzantine school, yet the beauty of the figures and the keen feeling for graceful curves and folds in the drapery point to a native Italian as being the artist who produced this wonderful work of art.

• James Gregory, in his Optica Promota (1663), discusses the forms of images and objects produced by lenses and mirrors, and shows that when the surfaces of the lenses or mirrors are portions of spheres the images are curves concave towards the objective, but if the curves of the surfaces are conic sections the spherical aberration is corrected.

• Consequently, for a certain focal length, much deeper curves must be resorted to if the new glasses are to be employed; this means not only greater difficulties in workmanship, but also greater thickness of glass, which militates against the chance of obtaining large disks quite free from striae and perfect in their state of annealing.

• Moreover the greater depths of the curves (or "curvature powers") in itself neutralize more or less the advantages obtained from the reduced irrationality of dispersion.

• In the paper which immediately follows, he gives the oft-quoted expression for the difference of slope (dp/d9) 8 -(dp/de) 1 of the vapour-pressure curves of a solid and liquid at the triple point, which is immediately deducible from (21), viz.

• For equilibrium, the altitude of the centre of gravity G must be stationary; hence G must lie in the same vertical line with the point of contact J of the two curves.

• If the body be supposed to roll - (say to the right) until the curves touch at J, and if JJ=bs, the angle through which the u,pper figure rotates is Is/p +Is/p, and the horizontal displace- V ment, of G is equal to the product of this expression into h.

• Such curves are often traced mechanically in acoustical and other experiments.

• If E be the point in which the line of the string meets AB, we have pjCP, pz=EP. Many contrivances for actually drawing the resulting curves have been devised.

• Cotes (1682-1716), and the various curves obtained are known as Cotess spirals.

• The question presents itself whether ther then is any other law of force, giving a finite velocity from infinity, under which all finite orbits are necessarily closed curves.

• If the velocity ratio is to be variable, as in s Williss Class B, the figures of the wheels are a pair of rolling curves, subject to the condition that the distance between their poles (which are the centres of rotation) shall be constant.

• A condition equivalent to the above, and necessarily connected with it, is, that at each pair of points of contact the inclinations of the curves to their radii-vectores shall be equal and contrary; or, denoting by r1, rf the radii-vectores at any given pair of points of contact, and s the length of the equal arcs measured from a certain fixed pair of points of contact dri/ds= drm/ds; (18)

• For full details as to rolling curves, see Williss work, already mentioned, and Clerk Maxwells paper on Rolling Curves, Trans.

• During the recess (in which the position of the teeth is exemplified in the figure by curves marked with accented letters), the face BiAi of the driving tooth drives the flank B2A2 of the following tooth, and the teeth are sliding from each other.

• Any other convenient figure may be assumed for the path of contact, and the corresponding forms of the teeth found by determining what curves a point T, moving along the assumed path of contact, will trace on two disks rotating round the centres of the wheels with angular velocities bearing that relation to the component velocity of T along TI, which is given by Principle II.

• The teeth of wheels of any figure, as well as of circular wheels, may be traced by rolling curves on their pitch-surfaces; and all teeth of the same pitch, traced by the same rolling curve with the same tracing-point, will work together correctly if their pitchsurfaces are in rolling contact.

• Trundles and Pin-Wheels.If a wheel or trundle have cylindrical pins or staves for teeth, the faces of the teeth of a wheel suitable for driving it are described by first tracing external epicycloids, by rolling the pitch-circle of the pin-wheel or trundle on the pitch-circle of the driving-wheel, with the centre of a stave for a tracing-point, and then drawing curves parallel to, and within the epicycloids, at a distance from them equal to the radius of a stave.

• The operations of describing the exact figures of the teeth of bevelwheels, whether by involutes or by rolling curves, are in every respect analogous to those for describing the figures of the teeth of spur-wheels, except that in the case of bevel-wheels all those operations are to be performed on the surface of a sphere described about the apex instead of on a plane, substituting poles for centres, and o B2

• To the north-west of the volcanic island of Zebayir the depth is less than 500 fathoms; the bottom of the channel rises to the ioofathom line at Hanish Island (also volcanic), then shoals to 45 fathoms, and sinks again in about the latitude of Mokha in a narrow channel which curves westward round the island of Perim (depth 170 fathoms), to lose itself in the Indian Ocean.

• In July, on the other hand, the isotherms show an almost constant temperature all over the country, and the linguiform curves are wanting.

• North of Bhutan, between the Himalayan crest and Lhasa, this formation is approximately maintained; farther east, although the same natural forces first resulted in the same effect of successive folds of the earth's crust, forming extensive curves of ridge and furrow, the abundant rainfall and the totally distinct climatic conditions which govern the processes of denudation subsequently led to the erosion of deeper valleys enclosed between forest-covered ranges which rise steeply from the river banks.

• The railroad in making this ascent makes curves equivalent to forty-two whole circles in a distance of 81 m., at one place paralleling its track five times in a space of about 300 ft.

• Also, since the curves at P and p are equally inclined to the directrix, P and p are corresponding, points and the line P p must pass through the centre of similitude.

• In the annexed figure, there are shown various examples of the curves named above, when the radii of the rolling and fixed circles are in the ratio of I to 3.

• Similarly, the corresponding epitrochoids will exhibit three loops or nodes (curve b), or assume the form shown in the curve; c. It is interesting to compare the forms of these curves with the three forms of the cycloid.

• The equations to the hypocycloid and its corresponding trochoidal curves are derived from the two preceding equations by changing the sign of b.

• The epicychid when the radii of the circles are equal is the cardioid (q.v), and the corresponding trochoidal curves are limacons.

• The chief public buildings are the two Dutch Reformed churches, the old church being a good specimen of colonial Dutch architecture, with gables, curves and thatched roof.

• Among his most remarkable works may be mentioned his ten memoirs on quantics, commenced in 1854 and completed in 1878; his creation of the theory of matrices; his researches on the theory of groups; his memoir on abstract geometry, a subject which he created; his introduction into geometry of the "absolute"; his researches on the higher singularities of curves and surfaces; the classification of cubic curves; additions to the theories of rational transformation and correspondence; the theory of the twenty-seven lines that lie on a cubic surface; the theory of elliptic functions; the attraction of ellipsoids; the British Association Reports, 1857 and 1862, on recent progress in general and special theoretical dynamics, and on the secular acceleration of the moon's mean motion.

• In almost all Crustacea the food-canal runs straight through the body, except at its anterior end, where it curves downwards to the ventrally-placed mouth.

• As this arrangement extends also to the margins, the wings are more or less twisted upon themselves and present a certain degree of convexity on their superior or upper surface, and a corresponding concavity on their inferior or under surface, - their free edges supplying those fine curves which act with such efficacy upon the air in obtaining the maximum of resistance and the minimum of displacement.

• The twisting referred to is partly a vital and partly a mechanical act; - that is, it is occasioned in part by the action of the muscles and in part by the greater resistance experienced from the air by the tip and posterior margin of the wing as compared with the root and anterior margin, - the resistance experienced by the tip and posterior margin causing them to reverse always subsequently to the root and anterior margin, which has the effect of throwing the anterior and posterior margins of the wing into figure-of-8 curves, as shown at figs.

• The midland plain curves northward between the outcrop of the Dolomite on the west and the Oolitic heights on the east.

• The escarpment runs north from Portland Island on the English Channel, curves north-eastward as the Cotteswold Hills, rising abruptly from the Severn plain to heights of over Iwo ft.; it sinks to insignificance in the Midland counties, is again clearly marked in Lincolnshire, and rises in the North Yorkshire moors to its maximum height of over 1500 ft.

• The name cycloid is now restricted to the curve described when the tracing-point is on the circumference of the circle; if the point is either within or without the circle the curves are generally termed trochoids, but they are also known as the prolate and curtate cycloids respectively.

• The cycloid is the simplest member of the class of curves known as roulettes.

• A historical bibliography of these curves is given in Brocard, Notes de bibliographie des courbes geometriques (1897).

• In optics, the term caustic is given to the envelope of luminous rays after reflection or refraction; in the first case the envelope is termed a catacaustic, in the second a diacaustic. Catacaustics are to be observed as bright curves when light is allowed to fall upon a polished riband of steel, such as a watch-spring, placed on a table, and by varying the form of the spring and moving the source of light, a variety of patterns may be obtained.

• 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/(2n4-I), where a is the radius of the mirror and n the number of reflections.

• 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.

• 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.

• In the first volume Of the Entwickelungen he applied the method of abridged notation to the straight line, circle and conic sections, and he subsequently used it with great effect in many of his researches, notably in his theory of cubic curves.

• Cramer, that, when a certain number of the intersections of two algebraical curves are given, the rest are thereby determined.

• Gergonne had shown that when a number of the intersections of two curves of the (p+q)th degree lie on a curve of the pth degree the rest lie on a curve of the qth degree.

• Plucker finally (Gergonne Ann., 1828-1829) showed how many points must be taken on a curve of any degree so that curves of the same degree (infinite in number) may be drawn through them, and proved that all the points, beyond the given ones, in which these curves intersect the given one are fixed by the original choice.

• Jacobi, he extended these results to curves and surfaces of unequal order.

• His discussion of curves of the third order turned mainly on the nature of their asymptotes, and depended on the fact that the equation to every such curve can be put into the form pqr-hus = o.

• The work falls into two parts, which treat of the asymptotes and singularities of algebraical curves respectively; and extensive use is made of the method of counting constants which plays so large a part in modern geometrical researches.

• Following the curves of its axis from west to east the lake is about 383 m.

• For simplicity of calculation Rankine chose logarithmic curves for both the inner and outer faces, and they fit very well with the conditions.

• In a small commonplace book, bearing on the seventh page the date of January 1663/1664, there are several articles on angular sections, and the squaring of curves and " crooked lines that may be squared," several calculations about musical notes, geometrical propositions from Francis Vieta and Frans van Schooten, annotations out of Wallis's Arithmetic of Infinities, together with observations on refraction, on the grinding of " spherical optic glasses," on the errors of lenses and the method of rectifying them, and on the extraction of all kinds of roots, particularly those " in affected powers."

• Halley only communicated to Newton the fact " that Hooke had some pretensions to the invention of the rule for the decrease of gravity being reciprocally as the squares of the distances from the centre," acknowledging at the same time that, though Newton had the notion from him, " yet the demonstration of the curves generated thereby belonged wholly to Newton."

• The first contains an explanation of the doctrine of fluxions, and of its application to the quadrature of curves; the second, a classification of seventy-two curves of the third order, with an account of their properties.

• The problem was to find the orthogonal trajectories of a series of curves represented by a single equation.

• The observer cannot long continue his researches in the field without discovering that the rocks of the earth's crust have been almost everywhere thrown into curves, usually so broad and gentle as to escape observation except when specially looked for.

• The outcrop of beds at the surface is commonly the truncation of these curves.

• It is one of the most crooked streams in the world, and its length in a straight line is less than half that by its curves.

• On the south side, above the Xingu, a line of low bluffs extends, in a series of gentle curves with hardly any breaks nearly to Santarem, but a considerable distance inland, bordering the flood-plain, which is many miles wide.

• The Greek geometers invented other curves; in particular, the conchoid, which is the locus of a point such that its distance from a given line, measured along the line drawn through it to a fixed point, is constant; and the cissoid, which is the locus of a point such that its distance from a fixed point is always equal to the intercept (on the line through the fixed point) between a circle passing through the fixed point and the tangent to the circle at the point opposite to the fixed point.

• Considering an equation in point-co-ordinates, we may have among the component curves right lines, and if in order to put these in evidence we take the equation to be L 1 Y1..

• We hence divide curves according to their order, viz.

• But it is an improper quadric curve; and in speaking of curves of the second or any other given order, we frequently imply that the curve is a.

• The intersections of two curves are obtained by combining their equations; viz.

• Supposing that the two curves are of the orders m, n, respectively, then the order of the resultant equation is in general and at most = mn; in particular, if the curve of the order n is an arbitrary line (n= 1), then the order of the resultant equation is = m; and the curve of the order m meets therefore the line in m points.

• Descartes in the Geometric defined and considered the remarkable curves called of ter him the ovals of Descartes, or simply Cartesians, which will be again referred to.

• Various properties of curves in general, and of cubic curves, are established in Colin Maclaurin's memoir, "De linearum geometricarum proprietatibus generalibus Tractatus " (posthumous, say 1746, published in the 6th edition of his Algebra).

• And we thus see how the theorem extends to curves, their points and tangents; if there is in the first figure a curve of the order m, any line meets it in m points; and hence from the corresponding point in the second figure there must be to the corresponding curve m tangents; that is, the corresponding curve must be of the class in.

• We may further consider the inflections and double tangents, as well in general as in regard to cubic and quartic curves.

• A better process was indicated by Salmon in the " Note on the Double Tangents to Plane Curves," Phil.

• The solution is still in so far incomplete that we have no properties of the curve II = o, to distinguish one such curve from the several other curves which pass through the points of contact of the double tangents.

• They are the polar curves of the point with regard to u= o.

• The notion is very probably older, but it is at any rate to be found in Lagrange's Theorie des fonctions analytiques (1798); it is there remarked that the equation obtained by the elimination of the parameter a from an equationf (x,y,a) = o and the derived equation in respect to a is a curve, the envelope of the series of curves represented by the equation f (x,y,a) = o in question.

• Similarly among the common tangents of the two curves we have the double tangents each counting twice, and the stationary tangents each counting three times, and the number of the remaining common tangents is = n 2 - 27-- 3e (=m 2 -26-3K, inasmuch as each of these numbers is as was seen = m+n).

• At any one of the m 2 -26 - 3K points the variable curve and the consecutive curve have tangents distinct from yet infinitesimally near to each other, and each of these two tangents is also infinitesimally near to one of the n 2 -2T-3t common tangents of the two curves; whence, attending only to the variable curve, and considering the consecutive curve as coming into actual coincidence with it, the n 2 -2T-3c common tangents are the tangents to the variable curve at the m 2 -26-3K points respectively, and the envelope is at the same time generated by the m 2 -26-3K points, and enveloped by the n2-2T-3c tangents; we have thus a dual generation of the envelope, which only differs from Pliicker's dual generation, in that in place of a single point and tangent we have the group of m2-26-3K points and n 2 -2T-3c tangents.

• The parameter which determines the variable curve may be given as a point upon a given curve, or say as a parametric point; that is, to the different positions of the parametric point on the given curve correspond the different variable curves, and the nature of the envelope will thus depend on that of the given curve; we have thus the envelope as a derivative curve of the given curve.

• Many well-known derivative curves present themselves in this manner; thus the variable curve may be the normal (or line at right angles to the tangent) at any point of the given curve; the intersection of the consecutive normals is the centre of curvature; and we have the evolute as at once the locus of the centre of curvature and the envelope of the normal.

• It may be added that the given curve is one of a series of curves, each cutting the several normals at right angles.

• Considering always real curves, we obtain the notion of a branch; any portion capable of description by the continuous motion of a point is a branch; and a curve consists of one or more branches.

• Thus the curve of the first order or right line consists of one branch; but in curves of the second order, or conics, the ellipse and the parabola consist each of one branch, the hyperbola of two branches.

• The five divergent parabolas are curves each of them symmetrical with regard to an axis.

• The author considers not only plane curves, but also cones, or, what is almost the same thing, the spherical curves which are their sections by a concentric sphere.

• And it then appears that there are two kinds of non-singular cubic cones, viz, the simplex, consisting of a single sheet, and the complex, consisting of a single sheet and a twin-pair sheet; and we thence obtain (as for cubic curves) the crunodal, the acnodal and the cuspidal kinds of cubic cones.

• The circular cubic and the bicircular quartic, together with the Cartesian (being in one point of view a particular case thereof), are interesting curves which have been much studied, generally, and in reference to their focal properties.

• The notions of distance and of lines at right angles are connected with the circular points; and almost every construction of a curve by means of lines of a determinate length, or at right angles to each other, and (as such) mechanical constructions by means of linkwork, give rise to curves passing the same definite number of times through the two circular points respectively, or say to circular curves, and in which the fixed centres of the construction present themselves as ordinary, or as singular, foci.

• Again, the normal, qua line at right angles to the tangent, is connected with the circular points, and these accordingly present themselves in the before-mentioned theories of evolutes and parallel curves.

• The theory first referred to, with the resulting notion of " Geschlecht," or deficiency, is more than the other two an essential part of the theory of curves, but they will all be considered.

• The general theorem is that two curves corresponding rationally to each other have the same deficiency.

• The theorem of united points in regard to points in a right line was given in a paper, June-July 1864, and it was extended to unicursal curves in a paper of the same series (March 1866), " Sur les courbes planes ou a double courbure dont les points peuvent se determiner individuellement - application du principe de correspondance dans la theorie de ces courbes."

• The extension to curves of any given deficiency D was made in the memoir of Cayley, " On the correspondence of two points on a curve, " - Pore.

• Systems of Curves satisfying Conditions.

• Zeuthen in the case of curves of any given order establishes between the characteristics pc, v, and 18 other quantities, in all 20 quantities, a set of 24 equations (equivalent to 2 3 independent equations), involving(besides the 20 quantities) other quantities relating to the various forms of the degenerate curves, which supplementary terms he determines, partially for curves of any order, but completely only for quartic curves.

• It is the discussion and complete enumeration of the special or degenerate forms of the curves, and of the supplementary terms to which they give rise, that the great difficulty of the question seems to consist; it would appear that the 24 equations are a complete system, and that (subject to a proper determination of the supplementary terms) they contain the solution of the general problem.

• Regarding the ultimate curve as derived from a given penultimate curve, we connect with the ultimate curve, and consider as belonging to it, certain points called " summits " cn the component curves P 1 = o, P2 =o respectively; a summit / is a point such that, drawing from an arbitrary point 0 the tangents to the penultimate curve, we have OE as the limit of one of these tangents.

• It is a leading point in the theory that a curve in space cannot in general be represented by means of two equations U= o, V = o; the two equations represent surfaces, intersecting in a curve; but there are curves which are not the complete intersection of any two surfaces; thus we have the cubic in space, or skew cubic, which is the residual intersection of two quadric surfaces which have a line in common; the equations U= o, V= o of the two quadric surfaces represent the cubic curve, not by itself, but together with the line.

• Upon these follow special methods of induction applicable to quantity, viz., the method of curves, the method of means, the method of least squares and the method of residues, and special methods depending on resemblance (to which the transition is made through the law of continuity), viz.

• Gunter's Quadrant, an instrument made of wood, brass or other substance, containing a kind of stereographic projection of the sphere on the plane of the equinoctial, the eye being supposed to be placed in one of the poles, so that the tropic, ecliptic, and horizon form the arcs of circles, but the hour circles are other curves, drawn by means of several altitudes of the sun for some particular latitude every year.

• The term cissoid has been given in modern times to curves generated in similar manner from other figures than the circle, and the form described above is distinguished as the cissoid of Diodes.

• A cissoid angle is the angle included between the concave sides of two intersecting curves; the convex sides include the sistroid angle.

• Before Newton the problem was that of devising empirical curves to formally represent the observed inequalities in the motion of the moon around the earth.

• It also contained numerous bends and sharp curves, sources of the greatest difficulty to navigation.

• These conditions define two systems of curves called respectively the principal curves of constant retardation and the principal lines of like polarization, these latter lines dividing the field into regions in which the intensity is alternately greater and less than the fundamental intensity.

• The determination of the curves of constant retardation depends upon expressing the retardation in terms of the optical constants of the crystal, the angle of incidence and the azimuth of the plane of incidence.

• Bertin has shown that a useful picture of the form of these curves may be obtained by taking sections, parallel to the plate, of a surface that he calls the "isochromatic surface," and that is the locus of points on the crystal at which the relative retardation of two plane waves passing simultaneously through a given point and travelling in the same direction has an assigned value.

• In the case of a biaxal plate perpendicular to the bisector of the acute angle between the optic axes, the curves of constant retardation are approximately Cassini's ovals, and the lines of like polarization are equilateral hyperbolae passing through the points corresponding to the optic axes.

• The isochromatic lines, unless the dispersion be excessive, follow in the main the course of the curves of constant retardation, and the principal lines of like polarization are with a crossed polarizer and analyser dark brushes, that in certain cases are fringed with colour.

• With a combination of plates in plane-polarized and plane-analysed light the interference pattern with monochromatic light is generally very complicated, the dark curves when polarizer and analyser are crossed being replaced by isolated dark spots or segments of lines.

• Twice in the year, he observed, they seem to travel across the solar disk in straight lines; at other times, in curves.

• These appearances he referred with great acuteness to the slight inclination of the sun's axis of rotation to the plane of the ecliptic. Thus, when the earth finds herself in the plane of the sun's equator, which occurs at two opposite points of her orbit, the spots, travelling in circles parallel with that plane, necessarily appear to describe right lines; but when the earth is above or below the equatorial level, the paths of the spots open out into curves turned downwards or upwards, according to the direction in which they are seen.

• In accordance with this hypothesis, the curves representing the variations of thermoelectric power, dE/dt, with temperature 'OObservationsof' Pia.

• It should also be remarked that even if the curves were not parabolas, it would always be possible to draw parabolas to agree closely with the observations over a restricted range of temperature.

• Its duct leaves the inferior anterior angle, at first descends a little, and runs forward under cover of the rounded inferior border of the lower jaw, then curves up along the anterior margin of the masseter muscle, becoming superficial, pierces the buccinator, and enters the mouth by a simple aperture opposite the middle of the crown of the third premolar tooth.

• Owing to the varying latitude of the ship, and the fact that the observer attempted to draw curves of equal brilliancy instead of the central line, the required conclusions cannot be drawn with certainty from these observations.

• The definitions given above reflect the intimate association of these curves, but it frequently happens that a particular conic is defined by some special property (as the ellipse, which is the locus of a point such that the sum of its distances from two fixed points is constant); such definitions and other special properties are treated in the articles Ellipse, Hyperbola and Parabola.

• The invention of the conic sections is to be assigned to the school of geometers founded by Plato at Athens about the 4th century B.C. Under the guidance and inspiration of this philosopher much attention was given to the geometry of solids, and it is probable that while investigating the cone, Menaechrnus, an associate of Plato, pupil of Eudoxus, and brother of Dinostratus (the inventor of the quadratrix), discovered and investigated the various curves made by truncating a cone.

• That he made considerable progress in the study of these curves is evidenced by Eutocius, who flourished about the 6th century A.D., and who assigns to Menaechmus two solutions of the problem of duplicating the cube by means of intersecting conics.

• Archimedes contributed to the knowledge of these curves by determining the area of the parabola, giving both a geometrical and a mechanical solution, and also by evaluating the ratio of elliptic to circular spaces.

• Pappus in his commentary on Apollonius states that these names were given in virtue of the above relations; but according to Eutocius the curves were named the parabola, ellipse or hyperbola, according as the angle of the cone was equal to, less than, or greater than a right angle.

• John Wallis, in addition to translating the Conics of Apollonius, published in 1655 an original work entitled De sectionibus conicis nova methodo expositis, in which he treated the curves by the Cartesian method, and derived their properties from the definition in piano, completely ignoring the connexion between the conic sections and a cone.

• A method of generating conics essentially the same as our modern method of homographic pencils was discussed by Jan de Witt in his Elementa linearum curvarum (1650); but he treated the curves by the Cartesian method, and not synthetically.

• Eagles, Constructive Geometry of Plane Curves (1886); geometric investigations primarily based on the relation of the conic sections to a cone are given in Hugo Hamilton's De Sectionibus Conicis (1758); this method of treatment has been largely replaced by considering the curves from their definition in piano, and then passing to their derivation from the cone and cylinder.

• The most important are those relating to algebraical curves and surfaces, especially the short paper Allgemeine Eigenschaften algebraischer Curven.

• In the same volume are treatises on "Geometric Loci, or Spherical Tangencies," and on the "Rectification of Curves," besides a restoration of "Apollonius's Plane Loci," together with the author's correspondence addressed to Descartes, Pascal, Roberval, Huygens and others.

• In the Duddell oscillograph it is usual to place a pair of loops in the magnetic field, each with its own mirror, so that a pair of curves can be delineated at the same time, and if there is any difference in phase between them, it will be detected.

• Thus we can take two curves, one showing the potential difference at the end of an inductive circuit, and the other the current flowing through the circuit.

• Its formation is characteristic, consisting of a series of ridges forming a succession of curves from a common centre.

• The soft curves on that tall slender body and those long legs were the talk of the locker room when he was a senior.

• Katie moved toward them steadily, self-conscious in the snug teal gown that displayed the curves the slender women around her didn't have.

• It wasn't as if she were wearing a bikini, and her only physical attributes were a flat abdomen and smooth curves â€“ well, those and her breasts, but they were over proportioned - out of balance, so to speak.

• She wore a shift sheer enough for him to see the shading of her curves.

• Her body was warm beneath his, a sensual combination of firm muscle beneath soft, feminine curves.

• Adjacent pixels creates smooth, natural curves.

• It is called algebraic Curves over a Finite Field and is currently 644 pages.

• After the sharply angular esthetic of the exterior of the church the visitor is surprised by the curves that everywhere meet the eye inside.

• With a higher backrest and subtle curves for your body, it may be difficult to coax anybody out of it!

• The Danube curves gently and reflects in Budapest's broad avenues, leafy parks and elaborate bathhouses.

• It curves majestic canyons, yet ripples through the towns.

• Elliptic curves offer major advances on older systems such as increased speed, less memory and smaller key sizes.

• Oulton is a wide circuit, with long sweeping curves, a rare mixture of gradients and some picturesque scenery.

• Summary The shape of indifference curves depends upon the preferences of the individual.

• The chain rule enables you to calculate higher derivatives for these parametrically described curves.

• The partial pressure of carbon dioxide has a much smaller effect on the current voltage curves than the partial pressure of oxygen.

• The main analysis developments will be to develop methods to extract information about phonon dispersion curves.

• This new drivetrain creates a shorter front overhang and a more rearward center of gravity giving the Bengal better handling on curves.

• The unit's 4-band custom equalizer and preset EQ curves help you tailor the sound to your preference or environment.

• It is important to understand the origin of the graphs and curves displaying the excitation and emission spectra for a given fluorochrome.

• Emily Robinson, singer from the Dixie Chicks wore a form-fitting black dress that hugged her curves.

• It joins historic Sherwood Forest in the Northeast and curves round to Attenborough in the southwest.

• Points, lines, polygons, circles, arcs, and smooth curves can be freely intermixed with text.

• Good finish with thin laminations making an good choice for smooth curves as there are no hidden defects.

• I do remember that inquiry, and I always think of that as being probably one of my greatest learning curves there.

• Changes are made to the bikes fuel curves via the PC's onboard microprocessor.

• Good sitting posture maintains the spinal curves usually present in the erect standing position.

• Postscript is a Turing-complete reverse polish programming language with drawing primitives based on Bezier curves.

• The curves are a function of taper ratio and apply to straight-tapered wings.

• Mark out the shape of your pond using a thick rope or hosepipe for smooth curves.

• The Sanyo massage heads seesaw to an angle of 50 degrees to follow the curves of the shoulders and back.

• Ordinary matter curves space-time back on itself, like the surface of the Earth.

• Go trough and follow a partly sunken track that curves up to the right.

• Note how the common tangents from the free energy curves (upper graph) " construct " the phase diagram below it.

• Plot curves of coating thickness, t versus line speed, v for the three different pressures.

• The third pair of driving wheels had no flanges, to assist the long rigid wheelbase get round sharp curves.

• The Kew curves, for instance, might suggest that the range (maximum less minimum hourly value) was larger in June than in December.

• Here and there particular curves, for example, had been obliged to yield the secret of their tangent; but the ancient geometers apparently had no consciousness of the general bearings of the methods which they so successfully applied.

• When the directive force is constant, the curve is a cycloid; under other conditions, spirals and other curves are described (see Mechanics) .

• The decade from 1896 until 1905, inclusive, saw huge sums spent on yards, passing tracks, grade reduction, elimination of curves, substitution of large locomotives and cars for small ones, &c. During those ten years, the route mileage increased 34,991 m., or 17%, while the mileage of second, third, fourth and yard tracks and sidings increased 32,666 m., or nearly 57%.

• He will incline to the first of these alternatives when cheapness of first cost is a desideratum, but, except in unusually favourable circumstances, the resulting line, being full of sharp curves and severe gradients, will be unsuited for fast running and will be unable to accommodate heavy traffic economically.

• Reverse curves are compound curves in which the components are of contrary flexure, like the letter S; strictly the term is only applicable when the two portions follow directly one on the other, but it is sometimes used of cases in which they are separated by a " tangent " or portion of straight line.

• H.P. - (Were ?-W v r v)V ?2240WV ?2240WVa (22) 55 0 550 550G 550g where W e is weight of engine and tender in tons, Wv the weight of vehicles in tons, W the weight of train in tons =W e r e and r z, the respective engine and vehicle resistances taken from the curves fig.

• The committee had not found one that did not possess grave disadvantages, but concluded that the " principle of contact of the surfaces of vertical surfaces embodied in the Janney coupler afforded the best connexion for cars on curves and tangents "; and in 1887 the Association recommended the adoption of a coupler of the Janney type, which, as developed later, is shown in fig.

• The mouth is wide, bounded by stiff immobile lips, and curves slightly upwards at the hinder end.

• Just as the lines of flow of an electric current all pass in closed curves through the battery or other generator, so do all the lines of induction pass in closed curves through the magnet or magnetizing coil.

• The logarithmic curves which accompany his paper demonstrate that within wide ranges of maximum induction W = 0.01 B 1.6 = 0.5271 1 ' 62 very nearly.

• The curve showing the circumferential (or longitudinal) changes was also plotted, and from the two curves thus obtained it was easy, on the assumption that the metal was isotropic in directions at right angles to the magnetization, to calculate changes of volume; for if circumferential elongation be denoted by 1 1, and transverse elongation by 1 2, then the cubical dilatation (40r -) = l l 2/ 2 approximately.

• When such precautions are adopted it is found that the " electromotive force of magnetization " is, for a given specimen, perfectly definite both in direction and in magnitude; it is independent of the nature of the corrosive solution, and is a function of the field-strength alone, the curves showing the relation of electromotive force 'to field-intensity bearing a rough resemblance to the familiar I-H curves.

• Its northern extremity, Ras Musandan, rises precipitously from the straits of Hormuz; farther south the range curves inland somewhat, leaving a narrow but fertile strip, known as the Batina coast, between it and the sea, and containing several populous towns and villages of which Sohar, Barka and Sib are the chief.

• This work included the "Logometria," the trigonometrical theorem known as "Cotes' Theorem on the Circle" (see TRIGONOMETRY), his theorem on harmonic means, subsequently developed by Colin Maclaurin, and a discussion of the curves known as "Cotes' Spirals," which occur as the path of a particle described under the influence of a central force varying inversely as the cube of the distance.

• Any periodic curve may be resolved into sine or harmonic curves by Fourier's theorem.

• The Arithmetica infinitorum relates chiefly to the quadrature of curves by the so-called method of indivisibles established by Bonaventura Cavalieri in 1629 (see Infinitesimal Calculus).

• As our knowledge of the age and locality of weights increases these criteria in curves will prove of greater value; but even now no consideration of the connexion of different units should be made without a graphic representation to compare their relative extent and nature of variation.

• The group of two longnecked gazelles facing a palm tree is of extraordinary refinement, and shows the, artistic consciousness in every part; the symmetric rendering of the palm tree, reduced to fit the scale of the animals, the dainty grace of the smooth gazelles contrasted with the rugged stem, the delicacy of the long flowing curves and the fine indications of the joints, all show a sense of design which has rarely been equalled in the ceaseless repetitions of the tree and supporters motive during every age since.

• No other hand could have ventured to render the hair and beard of a sitter, as it was the habit of this inveterate linearist to do, not by indication of masses, but by means of an infinity of single lines swept, with a miraculous certainty and fineness of touch, in the richest amd most intricate of decorative curves.

• Such curves show differences of steepness according to the temperature (see temp. curve), and to alterations of light (lamp) and darkness.

• Allied to the matter just mentioned was Plucker's discovery of the six equations connecting the numbers of singularities in algebraical curves (see Curve).

• The divergent parabolas are of five species which respectively belong to and determine the five kinds of cubic curves; Newton gives (in two short paragraphs without any development) the remarkable theorem that the five divergent parabolas by their shadows generate and exhibit all the cubic curves.

• Menaechmus discussed three species of cones (distinguished by the magnitude of the vertical angle as obtuse-angled, right-angled and acuteangled), and the only section he treated was that made by a plane perpendicular to a generator of the cone; according to the species of the cone, he obtained the curves now known as the hyperbola, parabola and ellipse.

• The importance of this generalization cannot be overestimated; it is of more than historical interest, for it remains the basis upon which certain authorities introduce the study of these curves.

• I have often held in my hand a little model of the Plymouth Rock which a kind gentleman gave me at Pilgrim Hall, and I have fingered its curves, the split in the centre and the embossed figures "1620," and turned over in my mind all that I knew about the wonderful story of the Pilgrims.

• My soul delights in the repose and gracious curves of the Venus; and in Barre's bronzes the secrets of the jungle are revealed to me.

• I should think the wonderful rhythmical flow of lines and curves could be more subtly felt than seen.

• The little fellow who whirls his "New York Flyer" round the nursery, making "horseshoe curves" undreamed of by less imaginative engineers, is concentrating his whole soul on his toy locomotive.

• How handsome the great sweeping curves in the edge of the ice, answering somewhat to those of the shore, but more regular!

• Now they drew close to the fox which began to dodge between the field in sharper and sharper curves, trailing its brush, when suddenly a strange white borzoi dashed in followed by a black one, and everything was in confusion; the borzois formed a star-shaped figure, scarcely swaying their bodies and with tails turned away from the center of the group.

• Use the reflectance curves illustrating the spectral response patterns of these two categories to help explain your answer.

• Individual study ROC plots were produced for studies with at least 10 years ' clinical follow-up and the area under the curves was compared.

• Will it make the most of those seductive curves I know lovers can't resist?

• He seeks to create a harmonious balance between strong powerful shapes and tactile sensuous curves.

• A standard bread knife is serrated on the wrong side for left-hand cutting and curves off to the right, making thick uneven wedges.

• Describe how the shapes of these curves are altered by changing the values of the parameters.

• Sinuous curves of silver which can be worn on the wrist, or higher up on the arm.

• Be able to construct solubility curves from experimental data.

• Gravity curves space-time, that is both space and time.

• Understand what Ro means, and how to interpret Life Tables and survivorship curves.

• Progress continues in developing new techniques to match image curves over arbitrary viewpoints for extracting symmetry axes and recognition.

• Wiggly curves can behave in an in between way, and this number can be anything between one and two.

• The student paid special attention to the curves of the contour lines when reproducing the teacher's artwork.

• Percentile curves are presented in 4 different growth curves.

• Weight gain charts have curves on them, and these curved lines are each representative of a certain percentile.