# Curve Sentence Examples

The sharp

**curve**arrived before her memory of it.His finger left her jaw and softly followed the

**curve**of her neck.The black dress she wore fit her like a second skin, outlining every

**curve**, dip and nook of her body.This

**curve**with the values reduced from metres to feet is reproduced below.His lips followed, softly pressing against the

**curve**of her neck.It was still light enough to see across the gorge when an opening in the trees allowed, but the long swing to the far end of valley was away from the direction the vehicle had driven and blocked from sight by the

**curve**of the canyon.The first

**curve**frightened the hell out of him and he knew the brake pressure necessary to slow him from this speed could not be engaged all the way down the mountain without overheating the tiny pads to the point of ineffectiveness.The amount of superelevation required to prevent derailment at a

**curve**can be calculated under perfect running conditions, given the radius of curvature, the weight of the vehicle, the height of the centre of gravity, the distance between the rails, and the speed; but great experience 1 See The Times Engineering Supplement (August 22, 1906), p. 265.Dean pulled out of the

**curve**, searching ahead for a glimpse of his quarry as he continued to hug the right side of the narrow roadway.As she rounded the

**curve**in the staircase, the room became silent.AdvertisementHe began first a short ascent, then a drop to a sharp

**curve**he nearly missed, causing him to reduce his speed further.If a be greater than b the

**curve**resembles fig.Go up the hill and watch for cars so you can warn anyone before they get to the

**curve**.She rounded a

**curve**and slammed on the breaks.The

**curve**for cobalt is a very remarkable one.AdvertisementBy the device of a hypsographic

**curve**co-ordinating the vertical relief and the areas of the earth's surface occupied by each zone of elevation, according to the system introduced by Supan, 2 Wagner showed his results graphically.For instance, if the

**curve**is of S-form, the point of danger is when the train enters the contra-flexure, and it is not an easy matter to assign the best superelevation at all points throughout the double bend.It is also the inverse of the same

**curve**for the same point.He grabbed the door handle as she spun around a

**curve**.It flashed, silver glinting off its graceful

**curve**.AdvertisementThis three-peaked

**curve**is not wholly pecuiiar to Paris, being seen, for instance, at Lisbon in summer.His reputation mainly rests on his Introduzione ad una teoria geometrica delle

**curve**piane, which proclaims him as a follower of the Steinerian or synthetical school of geometricians.If the asymptotes be perpendicular, or, in other words, the principal axes be equal, the

**curve**is called the rectangular hyperbola.When a train is running round a

**curve**the centrifugal force which comes into play tends to make its wheel-flanges press against the outer rail, or even to capsize it.The smoothest and safest running is, in fact, attained when a " transition," " easement " or " adjustment "

**curve**is inserted between the tangent and the point of circular curvature.AdvertisementThe 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.In this case the centre is a crunode and the

**curve**resembles fig.In this

**curve**ABCD are nodes.Hence the moment of the load on Am at C is wy0m, and the moment of a uniform load over any portion of the girder is w X the area of the influence

**curve**Ip' G' E ' under that portion.This method distorts the

**curve**, so that vertical ordinates of the**curve**are drawn to a scale b times greater than that of the horizontal ordinates.The vertical distortion of the

**curve**must not be so great that there is a very sensible difference between the length of the arc and its chord.The old Austria was very richly provided with raw materials; the coal and iron supply was especially rich; in the years immediately preceding the war the production of these two commodities followed in general a rising

**curve**.These ideas are further developed in various papers in the Bulletin and in his L'Anthropometrie, ou mesure des differentes facultes de l'homme (18'ji), in which he lays great stress on the universal applicability of the binomial law, - according to which the number of cases in which, for instance, a certain height occurs among a large number of individuals is represented by an ordinate of a

**curve**(the binomial) symmetrically situated with regard to the ordinate representing the mean result (average height).Thus the path of the ray when the aether is at rest is the

**curve**which makes fds/V least; but when it is in motion it is the**curve**which makes fds/(V+lug-m y -I-nw) least, where (l,m,n) is the direction vector of Ss.There will probably be a learning

**curve**in the arrival of wireless USB.The form of the torque

**curve**, or crank effort**curve**, as it is sometimes called, is discussed in the article Steam Engine, and the torque**curve**corresponding to actual indicator diagrams taken from an express passenger engine travelling at a speed of 65 m.Another of Roberval's discoveries was a very general method of drawing tangents, by considering a

**curve**as described by a moving point whose motion is the resultant of several simpler motions.It flows east and south in a wide

**curve**, through a broad upper valley past Chippenham and Melksham, after which it turns abruptly west to Bradford-on-Avon, receives the waters of the Frome from the south, and enters the beautiful narrow valley in which lie Bath and Bristol.The name lemniscate is sometimes given to any crunodal quartic

**curve**having only one real finite branch which is symmetric about the axis.The centre is a conjugate point (or acnode) and the

**curve**resembles fig.The instrument can be provided with a

**curve**or table showing the current corresponding to each angular displacement of the torsion head.If the attraction of a central body is not the only force acting on the moving body, the orbit will deviate from the form of a conic section in a degree depending on the amount of the extraneous force; and the

**curve**described may not be a re-entering**curve**at all, but one winding around so as to form an indefinite succession of spires.Let the

**curve**represent an elliptic orbit, AB being the major axis, DE the minor axis, and F the focus in which the centre of attraction is situated, which centre we shall call the sun.For the subjects under this heading see the articles CONIC SECTIONS; CIRCLE;

**CURVE**; GEOMETRICAL CONTINUITY; GEOMETRY, Axioms of; GEOMETRY, Euclidean; GEOMETRY, Projective; GEOMETRY, Analytical; GEOMETRY, Line; KNOTS, MATHEMATICAL THEORY OF; MENSURATION; MODELS; PROJECTION; Surface; Trigonometry.The downward course of the

**curve**is, owing to hysteresis, strikingly different from its upward course, and when the magnetizing force has been reduced to zero, there is still remaining an induction of 7500 units.The effect produced by a current is exactly opposite to that of tension, raising the elongation

**curve**instead of depressing it.Some experiments were next undertaken with the view of ascertaining how far magnetic changes of length in iron were dependent upon the hardness of the metal, and the unexpected result was arrived at that softening produces the same effect as tensile stress; it depresses the elongation

**curve**, diminishing the maximum extension, and reducing the " critical value " of the magnetizing force.The experiments were not sufficiently numerous to indicate whether, as is possible, there is a critical degree of hardness for which the height of the elongation

**curve**is a maximum.Among other things, it was found that the behaviour of cast cobalt was entirely changed by annealing; the sinuous

**curve**shown in Fig.Ewing's independent experiments showed that the magnetization

**curve**for a cobalt rod under a load of 16.2 kilogrammes per square mm.When the

**curve**after its steep descent has almost reached the axis, it bends aside sharply and becomes a nearly horizontal straight line; the authors suggest that the critical temperature should be defined as that corresponding to the point of maximum curvature.The magnetization

**curve**was found to be of the same general form as that of a paramagnetic metal, and gave indications that with a sufficient force magnetic saturation would probably be attained.Rhoads obtained a cyclic

**curve**for iron which indicated thermo-electric hysteresis of the kind exhibited by Nagaoka's curves for magnetic strain.He also experimented with nickel and again found a resemblance to the strain

**curve**.The origin of co-ordinates 0 corresponds to v = 0; and the asymptotic points J, J', round which the

**curve**revolves in an ever-closing spiral, correspond to v= =co .For the osculating circle at any point includes the whole of the y

**curve**which lies beyond; and the successive convolutions envelop one another without intersection.The utility of the

**curve**depends upon the fact that the elements of arc represent, in amplitude and phase, the component vibrations due to the corresponding portions of the primary wave-front.If the slit is of 'constant width and we require the illumination at various points on the screen behind it, we must regard the arc of the

**curve**as of constant length.Yet another fashion was that adopted by the flamens, who passed the right-hand portion of the toga over the right shoulder and arm and back over the left shoulder, so that it hung down in a

**curve**over the front of the body; the upper edge was folded over.If a point be in motion in any orbit and with any velocity, and if, at each instant, a line be drawn from a fixed point parallel and equal to the velocity of the moving point at that instant, the extremities of these lines will lie on a

**curve**called the hodograph.At first it was known by the Dutch simply as the "fuyck" (hoop), from the

**curve**in the river at this point, whence was soon derived the name Beverfuyck or Beverwyck.They then form part of a system of ranges which

**curve**north of the sources of the Chindwin river, and with the Kumon range and the hills of the Jade and Amber mines, make up a highland tract separated from the great Northern Shan plateau by the gorges of the Irrawaddy river.West of this line the rocks are chiefly Tertiary and Quaternary; east of it they are mostly Palaeozoic or gneissic. In the western mountain ranges the beds are thrown into a series of folds which form a gentle

**curve**running from south to north with its convexity facing westward.The resultant horizontal thrust in any direction is obtained by drawing parallel horizontal lines round the boundary, and intersecting a plane perpendicular to their direction in a plane

**curve**; and then investigating the thrust on this plane area, which will be the same as on the curved surface.Then 4, =o over the cylinder r = a, which may be considered a fixed post; and a stream line past it along which 4, = Uc, a constant, is the

**curve**(r - ¢2) sin 0=c, (x2 + y2) (y - c) - a 2 y = o, (3) a cubic**curve**(C3).A relative stream line, along which 1/,' = Uc, is the quartic

**curve**y-c=?![2a(r-x)], x = 4a2y2-(y g)4, r- 4a2y2 +(y c) 4, 7) 4 a (y-c) 4a(y and in the absolute space**curve**given by 1', dy= (y- c)2, x= 2ac_ 2a log (y -c) (8) 2ay y - c 34.Dinostratus, a Greek geometer and disciple of Plato, discussed the

**curve**, and showed how it effected a mechanical solution of squaring the circle.From every point of the

**curve**of intersection, perpendiculars are drawn to the axis.The intercept on the axis of y is 2a/7r; therefore, if it were possible to accurately construct the

**curve**, the quadrature of the circle would be effected.The

**curve**also permits the solution of the problems of duplicating a cube and trisecting an angle.The vapour pressure-composition

**curve**will now be concave to the axis of composition, the minima corresponding to the pure components.The vapour tension may approximate to a linear function of the composition, and the

**curve**will then be practically a straight line.If we determine the freezing-points of a number of mixtures varying in composition from pure A to pure B, we can plot the freezing-point

**curve**.In such a

**curve**the percentage composition can be plotted horizontally and the temperature of the freezing-point vertically, as in fig.In the case of two substances which neither form compounds nor dissolve each other in the solid state, the complete freezing-point

**curve**takes the form shown infig.5.The freezingpoint

**curve**sometimes indicates the existence of chemical compounds.Sometimes a freezing-point

**curve**contains more than one intermediate summit, so that more than one compound is indicated.For example, in the

**curve**for gold-aluminium, ignoring minor singularities, we find two intermediate summits, one at the percentage Au 2 A1, and another at the percentage AuAl 2.In the

**curve**for sodium-cadmium, the compound NaCd 2 is plainly shown.We must not take it for granted, when the freezing-point

**curve**gives no indication of the compound, that the compound does not exist in the solid alloy.For example, the compound Cu3Sn is not indicated in the freezing-point

**curve**, and indeed a liquid alloy of this percentage does not begin to solidify by the formation of crystals of Cu 3 Sn; the liquid solidifies completely to a uniform solid solution, and only at a lower temperature does this change into crystals of the compound, the transformation being accompanied by a considerable evolution of heat.It is evident that any other property can be represented by similar diagrams. For example, we can construct the

**curve**of conductivity of alloys of two metals or the surface of conductivity of ternary alloys, and so on for any measurable property.This may be interpreted as the equation of the border

**curve**giving the relation between p and 0, but is more easily obtained by considering the equilibrium at constant pressure instead of constant volume.To find the border

**curve**of equilibrium between the two states, giving the saturation pressure as a function of the temperature, we have merely to equate the values of G and G".Two of these figures stood at the end of a re-entrant

**curve**, several pieces of which are preserved.As an additional claim to remembrance, he was the first to solve Leibnitz's problem of the isochronous

**curve**(Acta Eruditorum, 1690).He proposed the problem of the catenary or

**curve**formed by a chain suspended by its two extremities, accepted Leibnitz's construction of the**curve**and solved more complicated problems relating to it.He determined the "elastic

**curve**," which is formed by an elastic plate or rod fixed at one end and bent by a weight applied to the other, and which he showed to be the same as the curvature of an impervious sail filled with a liquid (lintearia).Among these were the exponential calculus, and the

**curve**called by him the linea brachistochrona, or line of swiftest descent, which he was the first to determine, pointing out at the same time the relation which this**curve**bears to the path described by a ray of light passing through strata of variable density.Newton defined the diameter of a

**curve**of any order as the locus of the centres of the mean distances of the points of intersection of a system of parallel chords with the**curve**; this locus may be shown to be a straight line.The normal vertical distribution of temperature is illustrated in

**curve**A of fig.**Curve**B shows the typical distribution of temperature in an enclosed sea, in this case the Sulu Basin of the Malay Sea, where from the level of the barrier to the bottom the temperature remains uniform or homothermic.**Curve**C shows a typical summer condition in the polar seas, where layers of sea-water at different temperatures are superimposed, the arrangement from the surface to 200 fathoms is termed FIG.The tubs are usually formed into sets of from 2 to 12, the front one being coupled up by a short length of chain to a clamping hook formed of two jaws moulded to the

**curve**of the rope which are attached by the " run rider," as the driver accompanying the train is called.As for the marvels of Peru, the walls of the temple of the sun in Cuzco, with their circular form and

**curve**inward, from the ground upward, are most imposing.The tangent scale moved freely in a socket fixed to the gun; its lower end rested on one of the cams, cut to a correct

**curve**.Its white stone houses form a long

**curve**between the uplands of Salisbury Plain,which sweep away towards the north and east, and the tract of park and meadow land lying south and west.A plane figure bounded by a continuous

**curve**, or a solid figure bounded by a continuous surface, may generally be most conveniently regarded as generated by a straight line, or a plane area, moving in a fixed direction at right angles to itself, and changing as it moves.The application of Simpson's rule, for instance, to a plane figure implies certain assumptions as to the nature of the bounding

**curve**.Similarly, analytical plane geometry deals with the

**curve**described by a point moving in a particular way, while analytical plane mensuration deals with the figure generated by an ordinate moving so that its length varies in a particular manner depending on its position.The most simple case is that in which the trapezette tapers out in such a way that the

**curve**forming its top has very close contact, at its extremities, with the base; in other words, the differential coefficients u', u", u"',.According to Proclus, a man named Hippias, probably Hippias of Elis (c. 460 B.C.), trisected an angle with a mechanical

**curve**, named the quadratrix.We can represent waves of longitudinal displacement by a

**curve**, and this enables us to draw very important conclusions in a very simple way.If this is done for every point we obtain a continuous

**curve**Apbqcrd, which represents the displacement at every point at the given instant, though by a length at right angles to the actual displacement and on an arbitrary scale.In ordinary sound waves the displacement is very minute, perhaps of the order 105 cm., so that we multiply it perhaps by ioo,000 in forming the displacement

**curve**.If the waves are continuous and each of the same shape they form a " train," and the displacement

**curve**repeats itself.At J the displacement is forward, but since the

**curve**at Q is parallel to the axis the displacement is approximately the same for all the points close to J, and the air is neither extended nor compressed, but merely displaced bodily a distance represented by JQ.The figure shows that when the

**curve**of displacement slopes down in the direction of propagation there is compression, and the pressure is above the normal, and that when it slopes up there is extension, and the pressure is below the normal.If a wave travels on without alteration the travelling may be represented by pushing on the displacement

**curve**.The distribution of velocity then is represented by the dotted

**curve**and is forward when the**curve**is above the axis and Dackward when it is below.Generally, if any condition in the wave is carried forward unchanged with velocity U, the change of 4 at a given point in time dt is equal to the change of as we go back along the

**curve**a distance dx = Udt at the beginning of dt.When the value of dyldx is not very small E is no longer constant, but is rather greater in compression and rather less in extension than -yP. This can be seen by considering that the relation between p and is given by a

**curve**and not by a straight line.If both vibrate, the point describes a

**curve**which appears continuous through the persistence of the retinal impression.The simplest form of wave, so far as our sensation goes - that is, the one giving rise to a pure tone - is, we have every reason to suppose, one in which the displacement is represented by a harmonic

**curve**or a**curve**of sines, y=a sin m(x - e).The form of the

**curve**is evidently as represented in fig.The maximum height of the

**curve**HM =a is the amplitude.If we transfer 0 to A, e=o, and the

**curve**may be represented by y=a sin A x.If now the

**curve**moves along unchanged in form in the direction ABC with uniform velocity U, the epoch e =OA at any time t will be Ut, so that the value of y may be represented as 2 y=a sin T (x - Ut).The chief experimental basis for supposing that a train of longitudinal waves with displacement

**curve**of this kind arouses the sensation of a pure tone is that the more nearly a source is made to vibrate with a single simple harmonic motion, and therefore, presumably, the more nearly it sends out such a harmonic train, the more nearly does the note heard approximate to a single pure tone.Let it be represented by a displacement

**curve**Ahbkc. Its periodicity implies that after a certain distance the displacement**curve**exactly repeats itself.Let ABCD be drawn at such level that the areas above and below it are equal; then ABCD is the axis of the

**curve**.Since the

**curve**represents a longitudinal disturbance in air it is always continuous, at a finite distance from the axis, and with only one ordinate for each abscissa.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...Now we can see that two notes of the same pitch, but of different quality, or different form of displacement

**curve**, will, when thus analysed, break up into a series having the same harmonic wave-lengths; but they may differ as regards the members of the series present and their amplitudes and epochs.Now we may resolve these trains by Fourier's theorem into harmonics of wave-lengths X, 2X, 3A, &c., where X=2AB and the conditions as to the values of y can be shown to require that the harmonics shall all have nodes, coinciding with the nodes of the fundamental

**curve**.The ordinate of the

**curve**changes sign as we pass through a node, so that successive sections are moving always in opposite directions and have opposite displacements.The dotted

**curve**represents the superposition, which simply doubles each ordinate.The tangent to the displacement

**curve**is always parallel to the axis, that is, for a small distance the successive particles are always equally displaced, and therefore always occupy the same volume.If the plane does not contain the centre, the

**curve**of intersection is a "small circle," and the solid cut off is a "segment."The

**curve**of the main arch is a parabola.The lower portion of the figure is the

**curve**of bending moments under the leading load.Till W 1 has advanced a distance a only one load is on the girder, and the

**curve**A"F gives bending moments due to W 1 only; as W1 advances to a distance a+b, two loads are on the girder, and the**curve**FG gives moments due to W 1 and W2.The heavy continuous line gives the last-mentioned

**curve**for the reverse direction of passage of the loads.With short bridges it is best to draw the

**curve**of maximum bending moments for some assumed typical set of loads in the way just described, and to design the girder accordingly.But very great accuracy in drawing this

**curve**is unnecessary, because the rolling stock of railways varies so much that the precise magnitude and distribution of the loads which will pass over a bridge cannot be known.Now, except for very short bridges and very unequal loads, a parabola can be found which includes the

**curve**of maximum moments.This parabola is the

**curve**of maximum moments for a travelling load uniform per ft.Consider any other point F of the

**curve**, fig.In the same work Wallis obtained an expression for the length of the element of a

**curve**, which reduced the problem of rectification to that of quadrature.At Montrejeau it receives on the left the Neste, and encountering at this point the vast plateau of Lannemezan is forced to turn abruptly east, flowing in a wide

**curve**to Toulouse.It follows exactly the

**curve**of the mainland, and is continued into Panama, under the name of the Cordillera de Chiriqui.The interior

**curve**formed by the Gulf of Mexico is comparatively regular and has a coast-line of about 1400 m.The outer

**curve**facing the Pacific is less regular, is deeply broken by the Gulf of California, and has a coast-line of 4574 m., including that of the Gulf.The principal watershed is formed by the sierras of the state of Mexico, from which streams flow north-east to the Gulf of Mexico, northwest to the Pacific and south-west to the same coast below its great eastward

**curve**.The form of the limacon depends on the ratio of the two constants; if a be greater than b, the

**curve**lies entirely outside the circle; if a equals b, it is known as a cardioid; if a is less than b, the**curve**has a node within the circle; the particular case when b= 2a is known as the trisectrix.Some of these reverted breeds have developed horns of considerable size, although not showing that regularity of

**curve**distinctive of the wild race.The horns of the males are very large, and

**curve**round after the manner of the wild goat, with a tuft of hair between and in front.The swiftest current te,-ids, by reason of centrifugal force, to follow the outer side of every significant

**curve**in the channel; hence the concave bank, against which the rapid current sweeps, is worn away; thus any chance irregularity is exaggerated, and in time a series of large serpentines or meanders is developed,, the most-symmetrical examples at present being those near Greenville, Miss.This low range turns westward in a

**curve**through the Rattlesnake Mountains towards the high Wind River Mountains (Gannett Peak, 3,775 ft.), an anticlinal range within the body of the mountain system, with flanking strata rising well on the slopes.The upper canines especially are of great size, and

**curve**outwards, forwards and upwards.The

**curve**itself is sometimes termed the " circumference."Any line through the centre and terminated at both extremities by the

**curve**, e.g.The general equation to the circle in trilinear co-ordinates is readily deduced from the fact that the circle is the only

**curve**which intersects the line infinity in the circular points.Apollonius' genius takes its highest flight in Book v., where he treats of normals as minimum and maximum straight lines drawn from given points to the

**curve**(independently of tangent properties), discusses how many normals can be drawn from particular points, finds their feet by construction, and gives propositions determining the centre of curvature at any point and leading at once to the Cartesian equation of the evolute of any conic.Entering the department of Gers, the Adour receives the Arros on the right bank and begins to describe the large westward

**curve**which takes it through the department of Landes to the sea.The waterside streets, however, follow the

**curve**of the beach.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.Proposition 30 describes the construction of a

**curve**of double curvature called by Pappus the helix on a sphere; it is described by a point moving uniformly along the arc of a great circle, which itself turns about its diameter uniformly, the point describing a quadrant and the great circle a complete revolution in the same time.The area of the surface included between this

**curve**and its base is found - the first known instance of a quadrature of a curved surface.The great chain of volcanoes which runs through Sumatra and Java is continued eastwards into the Moluccas, and terminates in a hooklike

**curve**which passes through the Damar Islands to the Banda group. Outside this hook lies a concentric arc of non-volcanic islands, including Tenimber, the Lesser Kei Islands, Ceram and Buru; and beyond is still a third concentric arc extending from Taliabu to the Greater Kei Islands.On the whole it appears that the older rocks are found more particularly towards the interior of the

**curve**, and the newer rocks towards the exterior.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.Newton called attention to the fact that a falling body moves in a

**curve**, diverging slightly from the plumb-line vertical.If all the ice be melted, we pass along the vapour pressure

**curve**of water OA.If all the water be frozen, we have the vapour pressure

**curve**of ice OB; while, if the pressure be raised, so that all the vapour vanishes, we get the**curve**OC of equilibrium between the pressure and the freezing point of water.If the supply of ice fails first the temperature will rise, and, since solid salt remains, we pass along a

**curve**OA giving the relation between temperature and the vapour pressure of the saturated solution.If, on the other hand, the salt of the cryohydrate fails before the ice the water given by the continued fusion dilutes the solution, and we pass along the

**curve**OB which shows the freezing points of a series of solutions of constantly increasing dilution.Or, by increasing the pressure, we eliminate the vapour and obtain the

**curve**OF giving the relation between pressure, freezing point and composition when a saturated solution is in contact with ice and salt.Experiments on the relation between temperature and concentration are illustrated by projecting the

**curve**OA of fig.The pressure at each point should be that of the vapour, but since the solubility of a solid does not change much with pressure, measurements under the constant atmospheric pressure give a

**curve**practically identical with the theoretical one.When this process is complete the temperature rises, and we pass along a new

**curve**giving the equilibrium between anhydrous crystals, solution and vapour.In this way two temperature points are obtained in the investigation - the higher giving a point on the equilibrium

**curve**, the lower showing the non-variant point.Proceeding along the

**curve**in either direction, we come to a non-variant or eutectic 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.The experi mental

**curve**of solubility is shown in fig.At A we 66 have the freezing point of pure water, which is lowered by the gradual addition of 46 ferric chloride in the manner shown by the

**curve**AB.When the

**curve**BC is reached, Fe 2 C1 6 - 12H 2 0 separates out, and the solution solidifies.Further renewal of water will cause first liquefaction, as the

**curve**CD is passed, and then resolidification to Fe 2 C1 6.7H 2 0 when DE is cut.But at intermediate compositions we can only guess at the form of the energy-composition

**curve**, and the freezing point composition**curve**, deduced from it, will vary according to the supposition which we make.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 second and third figures, too, show that the presence of solid solutions may simulate the phenomena of chemical combination, where the

**curve**reaches a maximum, and of non-variant systems where we get a minimum.All available evidence, from the freezing point

**curve**and from other sources must be scrutinized before an opinion is pronounced.If the temperature at which this dense spontaneous shower of crystals is found be determined for different concentrations of solution, we can plot a "supersolubility

**curve**," which is found generally to run roughly parallel to the "solubility**curve**" of steady equilibrium between liquid and already existing solid.The residual liquid would thus become richer in B, and the tem perature and composition would pass along the

**curve**till E, the eutectic point, was reached.But, if no solid be present initially, or if the cooling be rapid, the liquid of composition x becomes supersaturated and may cool till the supersaturation

**curve**is reached at b, and a cloud of A crystals comes down.The conditions may then remain those of equilibrium along the

**curve**f E, but before reaching f the solution may become supersaturated with B and deposit B crystals spontaneously.Even this correction is not sufficient in solution of sugar, where the theoretical

**curve**II lies below the experimental observations.The apparently strange and inconvenient position of the Stadium relatively to the Altis was due simply to the necessity of obeying the conditions of the ground, here determined by the

**curve**of the loweslopes which bound the valley on the north.A neat rainguard made of sheet metal, to the same

**curve**as the body of the periscope and almost 8 inches long, is attached to the upper prism box by two spring straps.On the landward side, Braila has the shape of a crescent, the

**curve**of its outer streets following the line of the old fortifications, dismantled in 1829.The

**curve**is symmetrical about the line x = y, and consists of two infinite branches asymptotic to the line x+y+a = o and a loop in the first quadrant.It may be traced by giving m various values in the equations x=3am/ ('1-1-m 3 ),' y=3am2 (1-1-m 3), since by eliminating m between these relations the equation to the

**curve**is obtained.The area of the loop, which equals the area between the

**curve**and its asymptote, is 3a/2.Hence the main watershed extends eastwards, culminating in the Aiguille de Scolette (11,500 ft.), but makes a great

**curve**to the north-west and back to the south-east before rising in the Rochemelon (11,605 ft.), which may be considered as a re-entering angle in the great rampart by which Italy is guarded from its neighbours.If the two hands be placed flat upon the table, in the angle between the two books, and the cloth pushed towards the corner, it will at once be rucked up into a fold which will follow a

**curve**not unlike that of the Alps.The half-fan is a combination of the two forms, but as regards pruning does not materially differ from the horizontal, as two opposite side branches are produced in succession upwards till the space is filled, only they are not taken out so abruptly, but are allowed to rise at an acute angle and then to

**curve**into the horizontal line.Potatoes are cultivated in all the provinces, but especially in the Palatinate and in the Spessart district, which lies in the north-west within a

**curve**of the Main.The coast-line extends in a double

**curve**from south-west to northeast, and is formed by a row of sand dunes, 171 m.The Maas, whose course is almost parallel to that of the Rhine, follows in a wide

**curve**the general slope of the country, receiving the Roer, the Mark and the Aa.In many Basidiomycetes minute branches arise below the septa; their tips

**curve**over the outside of the latter, and fuse with the cell above just beyond it, forming a clamp-connexion.The city faces upon a

**curve**in the river bank forming what is called the Bay of Asuncion, and is built on a low sandy plain, rising to pretty hillsides overlooking the bay and the low, wooded country of the Chaco on the opposite shore.'The Bug, another right-hand tributary of the Vistula, describes a wide

**curve**concentric with those of the middle Vistula and the Narew, and separates the Polish governments of Lublin and Siedlce from the Russian governments of Volhynia and Grodno.In this simple case the temperature cycle at a depth x is a precisely similar

**curve**of the same period, but with the amplitude reduced in the proportion rn ', and the phase retarded by the fraction mx/27r of a cycle.The cycloid was a famous

**curve**in those days; it had been discussed by Galileo, Descartes, Fermat, Roberval and Torricelli, who had in turn exhausted their skill upon it.Pascal solved the hitherto refractory problem of the general quadrature of the cycloid, and proposed and solved a variety of others relating to the centre of gravity of the

**curve**and its segments, and to the volume and centre of gravity of solids of revolution generated in various ways by means of it.Those of the upper jaw are directed upwards from their bases, so that they never enter the mouth, but pierce the skin of the face, thus resembling horns rather than teeth; they

**curve**backwards, downwards, and finally often forwards again, almost or quite touching the forehead.Points on the same

**curve**are supposed to have the same average number of auroras in the year, and this average number is shown adjacent to the**curve**.Between the Shetlands and Iceland we cross the

**curve**of maximum frequency, and farther north the frequency diminishes.Greenland lies to the north of Fritz's

**curve**of maximum auroral frequency, and the suggestion has been made that the zone of maximum frequency expands to the south as sun-spots increase, and contracts again as they diminish, the number of auroras at a given station increasing or diminishing as the zone of maximum frequency approaches to or recedes from it.The

**curve**of the arch turns in slightly below the springing, giving a horse-shoe shape.There are many local irregularities, but the general direction is maintained as far as the southern extremity of Greece, where the folds show a tendency to

**curve**towards Crete.She invented and discussed the

**curve**known as the "witch of Agnesi" (q.v.) or versiera.The ordinate of the dotted

**curve**which contains its "centre of gravity" has, of course, for its abscissa the "mean" number of glands; the maximum ordinate of the**curve**is, however, at 2.98, or sensibly at 3 glands, showing what Pearson has called the "modal" number of glands, or the number occurring most frequently.Trans., A., 1893) that this frequency may be closely represented by the

**curve**whose equation is y = O.21 122 5 x-( 332 (7.3 2 53 - x) 3.142.The

**curve**, and the observations it represents, are drawn in fig.The form of the

**curve**is shown in fig.Then EG produced meets FP in a point on the

**curve**.The cartesian parabola is a cubic

**curve**which is also known as the trident of Newton on account of its three-pronged form.John Wallis utilized the intersections of this

**curve**with a right line to solve cubic equations, and Edmund Halley solved sextic equations with the aid of a circle.If two roots are imaginary the equation is y 2 =(x 2 +a 2) (x - b) and the

**curve**resembles the parabolic branch, as in the preceding case.Some of these are united to the mainland and to each other by jetties which

**curve**round so as to form the Port de Refuge, a haven available only in fair weather.Large flowing leaves of acanthus and other plants were beaten out with wonderful spirit and beauty of

**curve**.The length of India from north to south, and its greatest breadth from east to west, are both about 1900 m.; but the triangle tapers with a pear-shaped

**curve**to a point at Cape Comorin, its southern extremity.The sable and roan antelopes are distinguished from Oryx by the stout and thickly ringed horns rising vertically from a ridge over the eyes at an obtuse angle to the plane of the lower part of the face, and then sweeping backwards in a bold

**curve**.The Specific Heat Itself Can Be Deduced Only By Differentiating The

**Curve**Of Observation, Which Greatly Increases The Uncertainty.In both cases the rise to a maximum is more rapid than the decline to a minimum, and in fact some of the minor peculiarities of the sunspot

**curve**are closely imitated by the light-curves of variable stars.Either of the two is the all, as, for example, the law of the convexity of the

**curve**is the law of the**curve**and the law of its concavity.But by a most skilful manoeuvre Narses contrived to draw his lines into a

**curve**, so that his mounted archers on each flank could aim their arrows at the backs of the troops who formed the other side of the Alamannic wedge.The width of the photographic sheet which receives the spot of light reflected from the mirrors in the above instruments is generally so great that in the case of ordinary changes the

**curve**does-not go off the paper.Cady, Terrestrial Magnetism, 1904, 9, 69, describing a declination magnetograph in which the record is obtained by means of a pen acting on a moving strip of paper, so that the

**curve**can be consulted at all times to see whether a disturbance is in progress.The

**curve**shows two rises, one at the beginning of winter, and the other at the commencement of the monsoon, and at both these times the people are driven indoors.In the east a well-defined mountain system runs nearly parallel to the Black Sea coast from Batum to Sinope, forming a gentle

**curve**with its convexity facing southwards.For example, in the case of a particle lying on a smooth

**curve**, or on a smooth surface, if it be displaced along the**curve**, or on the surface, the virtual work of the normal component of the pressure may be ignored, since it is of the second order.It is assumed that the form can be sufficiently represented by a plane

**curve**, that the stress (tension) at any point P of the**curve**, between the two portions which meet there, is in the direction of the tangent at P, and that the forces on any linear element s must satisfy the conditions of equilibrium laid down in I.We will suppose in the first instance that the

**curve**is plane.Suppose, for example, that we have a light string stretched over a smooth

**curve**; and let Rs denote the normal pressure (outwards from the centre of curvature) on bs.Next suppose that the

**curve**is rough; and let Fas be the tangential force of friction on s.Again, take the case of a string under gravity, in contact with a smooth

**curve**in a vertical plane.This is the intrinsic equation of the

**curve**.It follows that the osculating plane of the

**curve**formed by the string must contain the normal to the surface, i.The relation between x and t in any particular case may be illustrated by means of a

**curve**constructed with I as abscissa and x as ordinate.This is called the

**curve**of positions or space-time**curve**; its gradient represents the velocity.A

**curve**with I as abscissa and u as ordinate is called the**curve**of velocities or velocity-time**curve**.The

**curve**of positions corresponding to (4) is a parabola, and that of velocities is a straight line.It may be noticed that if the scales of x and be properly adjusted, the

**curve**of positions in the present problem is the portion of a cycloid extending from a vertex to a cusp.Take, for example, the case of a particle moving on a smooth

**curve**in a vertical plane, under the action of gravity and the pressure R of the**curve**.In the case of the pendulum the tension of the string takes the place of the pressure of the

**curve**.Moreover, the case n=2 is the only one in which the critical orbit (27) can be regarded as the limiting form of a closed

**curve**.The herpolhode

**curve**in the fixed plane is obviously confined between two concentric circles which it alternately touches; it is not in general a re-entrant**curve**.The centres of pressure at the joints are also called centres of resistance, and the

**curve**passing through these points is called a line of resistance.A

**curve**tangential to all the sides of the polygon is the line of pressures.The path of P is a

**curve**of the kind called epitrochoids.To work with a wheel of any other figure, its section must be a rolling

**curve**, subject to the condition that the perpendicular distance from the pole or centre of the wheel to a straight line parallel to the direction of the motion of the rack shall be constant.If the same rolling

**curve**R, with the same tracing-point T, be rolled on the outside of any other pitch-circle, it will have the fare of a tooth suitable to work with the flank AT.In like manner, if either the same or any other rolling

**curve**be rolled the opposite way, on the outside of the pitch-circle BB, so that the tracing point T shall start from A, it will trace the face AT of a tooth suitable to work with a flank traced by rolling the same**curve**R with the same tracing-point T inside any other pitch-circle.The figure of the path of con tact is that traced on a fixed plane by the tracing-point, when the rolling

**curve**is rotated in such a manner as always to touch a fixed straight line EIE (or EIE, as the case may be) at a fixed point I (or I).This equation gives a relation between the inclination of the

**curve**to the horizon and the height above the level of the liquid.Resolving vertically we find that the weight of the liquid raised above the level must be equal to T(sin 0 2 - sin 01), and this is therefore equal to the area P 1 P 2 A 2 A 1 multiplied by gp. The form of the capillary surface is identical with that of the " elastic

**curve**," or the**curve**formed by a uniform spring originally straight, when its ends are acted on by equal and T 2 opposite forces applied either to the ends themselves or to solid pieces attached to them.Drawings of the different forms of the

**curve**may be found in Thomson and Tait's Natural Philosophy, vol.Let us first determine the nature of a

**curve**, such that if it is rolled on the axis its origin will trace out the meridian section of the bubble.Since at any instant the rolling

**curve**is rotating about the point of contact with the axis, the line drawn from this point of contact to the tracing point must be normal to the direction of motion of the tracing point.Hence if N is the point of contact, NP must be normal to the traced

**curve**.Also, since the axis is a tangent to the rolling

**curve**, the ordinate PR is the perpendicular from the tracing point P on the tangent.Hence the relation between the radius vector and the perpendicular on the tangent of the rolling

**curve**must be identical with the relation between the normal PN and the ordinate PR of the traced**curve**.If A 1, C 1 are the disks, so that the distance between them is less than irr, the

**curve**must be produced beyond the disks before it is at its mean distance from the axis.If, on the other hand, the disks are at A2 and C2, so that the distance between them is greater than 7rr, the

**curve**will reach its mean distance from the axis before it reaches the disks.Beyond them a road branches to the right, sweeping round in a broad

**curve**to the space in front of the temple of Apollo.The outer side of this

**curve**is bounded by a row of treasuries, similar to those found at Delphi and Olympia, and serving to house the more costly offerings of various islands or cities.The

**curve**obtained on joining the former points then brings out a number of facts, foremost among which are (1) that as long as the conditions remain constant the doubling periods - i.e.It was also shown that exposure to light, dilution or exhaustion of the food-media, the presence of traces of poisons or metabolic products check growth or even bring it to a standstill; and the death or injury of any single cell in the filamentous series shows its effect on the

**curve**by lengthening the doubling period, because its potential progeny have been put out of play.If we place the base of the filament in each case on a base line in the order of the successive times of observation recorded, and at distances apart proportional to the intervals of time (8.30, 10.0, 10.30, 11.40, and so on) and erect the straightened-out filaments, the proportional length of each of which is here given for each period, a line joining the tips of the filaments gives the

**curve**of growth.Since the circumference of a circle is proportional to its radius, it follows that if the ratio of the radii be commensurable, the

**curve**will consist of a finite number of cusps, and ultimately return into itself.In the particular case when the radii are in the ratio of I to 3 the epicycloid (

**curve**a) will consist of three cusps external to the circle and placed at equal distances along its circumference.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 hypocycloid derived from the same circles is shown as

**curve**d, and is seen to consist of three cusps arranged internally to the fixed circle; the corresponding hypotrochoid consists of a three-foil and is shown in**curve**e.This

**curve**is the envelope of a line of constant length, which moves so that its extremities are always on two fixed lines at right angles to each other, i.e.of the line xla+y//= I, with the condition a 2 + 1 3 2 = I/a, a constant.The strata are thrown into folds which run in the direction of the mountain ridges, forming a

**curve**with the convexity facing the south-east.If, again, the wing be suddenly elevated in a strictly vertical direction, as at c d, the wing as certainly darts upwards and forwards in a double

**curve**to e, thus converting the vertical up strokes into an upward, oblique, forward stroke.Plinlimmon (2468 ft.) is the highest of the hills, and forms a sort of hydrographic centre for the group, as from its eastern base the Severn and the Wye take their rise - the former describing a wide

**curve**to east and south, the latter forming a chord to the arc in its southward course.It appears as a series of rounded hills of no great elevation, running in a

**curve**from the mouth of the Axe to Flamborough Head, roughly parallel with the Oolitic escarpment.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.Francis Schooten (Commentary on Descartes) assigns the invention of the

**curve**to Rene Descartes and the first publication on this subject after Descartes to Marin Mersenne.Evangelista Torricelli, in the first regular dissertation on the cycloid (De dimensione cycloidis, an appendix to his De dimensione parabolae, 1644), states that his friend and tutor Galileo discovered the

**curve**about 1599.John Wallis discussed both the history and properties of the

**curve**in a tract De cycloide published at Oxford in 1659.Honore Fabri (Synopsis geometrica, 1669) treated of the

**curve**and enumerated many theorems concerning it.Galileo attempted the evaluation by weighing the

**curve**against the generating circle; this rough method gave only an approximate value, viz., a little less than thrice the generating circle.Blaise Pascal determined the area of the section made by any line parallel to the base and the volumes and centres of gravity of the solids generated by revolving the

**curve**about its axis and base.The evaluation of the area of the

**curve**had made Roberval famous in France, but Descartes considered that the value of his investigation had been grossly exaggerated; he declared the problem to be of an elementary nature and submitted a short and simple solution.The mechanical properties of the cycloid were investigated by Christiaan Huygens, who proved the

**curve**to be tautochronous.When measured R from the vertex the results may be expressed in the forms s= 4a sin 20 and s = (8ay); the total length of the

**curve**is 8a.The

**curve**is shown in fig.The cartesian equation, referred to the fixed diameter and the tangent at B as axes may be expressed in the forms x= a6, y=a(I -cos 0) and y-a=a sin (x/afir); the latter form shows that the locus is the harmonic

**curve**.This change of direction corresponds to a

**curve**in the line of volcanic fissures which have contributed their products to the building of the islands.The simplest case of a caustic

**curve**is when the reflecting surface is a circle, and the luminous rays emanate from a point on the circumference.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 limacon, and hence the primary caustic is the evolute of this

**curve**.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.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.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.Another Asiatic species is the great shou (C. affinis) of the Chumbi Valley, in which the antlers

**curve**forwards in a remarkable manner.Finally, we have the thamin, or Eld's deer, C. (R.) eldi, ranging from Burma to Siam, and characterized by the continuous

**curve**formed by the beam and the brow-tine of the antlers.The antlers are large and

**curve**forwards, giving off an upright snag near the base, and several vertical tines from the upper surface of the horizontal portion.From the lower part of a carpel are produced several laterally placed ovules, which become bright red or orange on ripening; the bright fleshy seeds, which in some species are as large as a goose's egg, and the tawny spreading carpels produce a pleasing combination of colour in the midst of the long dark-green fronds, which

**curve**gracefully upwards and outwards from the summit of the columnar stem.It will be seen that the maximum ordinates lie upon the

**curve**A9 = constant dotted in the figure, and so, as the temperature of the ideal body rises, the wave-length of most intense radiation shifts from the infra-red X towards the luminous part of the spectrum.The integrated emission of energy is given by the area of the outer smoothed

**curve**(4), and the conclusion from this one holograph is that the " solar constant " is 2.54 calories.The line n shows the factor by which the index of refraction of the transmitted vibration is multiplied, and the

**curve**p the intensity of the absorbed vibration for that wave-length.The innermost, and most lofty, chain of mountains follows a

**curve**almost identical with that of the coast at a general distance of 120 m.The most elevated portion of the innermost range, the Drakensberg follows the

**curve**of the coast from south to north-east.A navigable channel extends in an irregular

**curve**from the bay of Hoi-how (Hai-K`ow) in the north to Tan-chow on the west coast.We find on the left-hand scale of yield that the height of the ordinate drawn to the 50-inch mean rainfall

**curve**from 200,000 on the capacity scale, is 1457 gallons per day per acre; and the straight radial line, which cuts the point of intersection of the curved line and the co-ordinates, tells us that this reservoir will equalize the flow of the two driest consecutive years.If, on the other hand, water is suddenly drawn off from a cistern supplied through a ball-cock, the flow through the ball-cock will be recorded, and will be represented by a sudden rise to a maximum, followed by a gradual decrease as the ball rises and the cistern fills; the result being a

**curve**having its asymptote in the original horizontal line.They are directed outwards, and

**curve**in an open spiral, with the tips directed outwards.It is remarkable for the great size of the horns of the old rams and the wide open sweep of their

**curve**, so that the points stand boldly out on each side, far away from the animal's head, instead of curling round nearly in the same plane, as in most of the allied species.The spiral horns are low at the crown, with a clear space between the roots, and sweep in a wide

**curve**, sloping slightly backwards, and clear of the cheek.Without mentioning the speculations which had been made, he asked Newton what would be the

**curve**described by a planet round the sun on the assumption that the sun's force diminished as the square of the distance.In June 1696 Bernoulli addressed a letter to the mathematicians of Europe challenging them to solve two problems - (1) to determine the brachistochrone between two given points not in the same vertical line, (2) to determine a

**curve**such that, if a straight line drawn through a fixed point A meet it in two points P 1, P 2, then AP 1 m +AP 2 m will be constant.He announced that the

**curve**required in the first problem must be a cycloid, and he gave a method of determining it.The

**curve**consists of one branch entirely to the left of the line x= 2a and having the axis of y as an asymptote.The second method is to work out by slow and sure steps the lines of descent of the different families, orders, and classes, and so either to arrive at the ancestral form of each class, or to plot out the

**curve**of evolution, which may then legitimately be projected into "the dark backward and abysm of time."A

**curve**is a line, or continuous singly infinite system of points.We consider in the first instance, and chiefly, a plane

**curve**described according to a law.Such a

**curve**may be regarded geometrically as actually described, or kinematically as in the course of description by the motion of a point; in the former point of view, it is the locus of all the points which satisfy a given condition; in the latter, it is the locus of a point moving subject to a given condition.Thus the most simple and earliest known

**curve**, the circle, is the locus of all the points at a given distance from a fixed centre, or else the locus of a point moving so as to be always at a given distance from a fixed centre.In a machine of any kind, each point describes a

**curve**; a simple but important instance is the " three-bar**curve**," or locus of a point in or rigidly connected with a bar pivoted on to two other bars which rotate about fixed centres respectively.Every

**curve**thus arbitrarily defined has its own properties; and there was not any principle of classification.The idea was to represent any

**curve**whatever by means of a relation between the co-ordinates (x, y) of a point of the**curve**, or say to represent the**curve**by means of its equation.Observe that the distinctive feature is in the exclusive use of such determination of a

**curve**by means of its equation.The Greek geometers were perfectly familiar with the property of an ellipse which in the Cartesian notation is x 2 /a 2 +y 2 /b 2 =1, the equation of the

**curve**; but it was as one of a number of properties, and in no wise selected out of the others for the characteristic property of the**curve**.We obtain from the equation the notion of an algebraical as opposed to a transcendental

**curve**, viz.