Curve Sentence Examples

curve
  • The sharp curve arrived before her memory of it.

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  • His finger left her jaw and softly followed the curve of her neck.

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  • The black dress she wore fit her like a second skin, outlining every curve, dip and nook of her body.

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  • This curve with the values reduced from metres to feet is reproduced below.

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  • His lips followed, softly pressing against the curve of her neck.

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  • 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.

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  • 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.

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  • 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.

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  • 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.

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  • As she rounded the curve in the staircase, the room became silent.

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  • He began first a short ascent, then a drop to a sharp curve he nearly missed, causing him to reduce his speed further.

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  • If a be greater than b the curve resembles fig.

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

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  • She rounded a curve and slammed on the breaks.

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  • The curve for cobalt is a very remarkable one.

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  • By 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.

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  • 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.

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  • It is also the inverse of the same curve for the same point.

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  • He grabbed the door handle as she spun around a curve.

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  • It flashed, silver glinting off its graceful curve.

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  • This three-peaked curve is not wholly pecuiiar to Paris, being seen, for instance, at Lisbon in summer.

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  • 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.

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  • If the asymptotes be perpendicular, or, in other words, the principal axes be equal, the curve is called the rectangular hyperbola.

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  • 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.

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  • 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.

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  • 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.

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  • In this case the centre is a crunode and the curve resembles fig.

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  • In this curve ABCD are nodes.

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  • 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.

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  • 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.

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  • 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.

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  • 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.

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  • 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).

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  • 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.

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  • There will probably be a learning curve in the arrival of wireless USB.

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  • 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.

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  • 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.

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  • 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.

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  • The name lemniscate is sometimes given to any crunodal quartic curve having only one real finite branch which is symmetric about the axis.

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  • The centre is a conjugate point (or acnode) and the curve resembles fig.

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  • The instrument can be provided with a curve or table showing the current corresponding to each angular displacement of the torsion head.

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  • 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.

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  • 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.

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  • 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.

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  • 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.

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  • The effect produced by a current is exactly opposite to that of tension, raising the elongation curve instead of depressing it.

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  • 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.

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  • 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.

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  • Among other things, it was found that the behaviour of cast cobalt was entirely changed by annealing; the sinuous curve shown in Fig.

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  • Ewing's independent experiments showed that the magnetization curve for a cobalt rod under a load of 16.2 kilogrammes per square mm.

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  • 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.

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  • 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.

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

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  • He also experimented with nickel and again found a resemblance to the strain curve.

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  • 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 .

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  • 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.

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  • 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.

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  • 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.

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  • 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.

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  • 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.

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  • 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.

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  • 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.

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  • 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.

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  • 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.

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  • 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).

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  • 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.

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  • Dinostratus, a Greek geometer and disciple of Plato, discussed the curve, and showed how it effected a mechanical solution of squaring the circle.

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  • From every point of the curve of intersection, perpendiculars are drawn to the axis.

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  • 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.

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  • The curve also permits the solution of the problems of duplicating a cube and trisecting an angle.

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  • The vapour pressure-composition curve will now be concave to the axis of composition, the minima corresponding to the pure components.

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  • The vapour tension may approximate to a linear function of the composition, and the curve will then be practically a straight line.

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  • 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.

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  • In such a curve the percentage composition can be plotted horizontally and the temperature of the freezing-point vertically, as in fig.

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  • 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.

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  • The freezingpoint curve sometimes indicates the existence of chemical compounds.

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  • Sometimes a freezing-point curve contains more than one intermediate summit, so that more than one compound is indicated.

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  • 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.

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  • In the curve for sodium-cadmium, the compound NaCd 2 is plainly shown.

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  • 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.

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  • 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.

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  • 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.

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  • 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.

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  • 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".

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  • Two of these figures stood at the end of a re-entrant curve, several pieces of which are preserved.

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  • As an additional claim to remembrance, he was the first to solve Leibnitz's problem of the isochronous curve (Acta Eruditorum, 1690).

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  • 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.

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  • 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).

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  • 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.

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  • 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.

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  • The normal vertical distribution of temperature is illustrated in curve A of fig.

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  • 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.

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  • 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.

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  • 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.

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  • 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.

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  • 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.

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  • 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.

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  • The application of Simpson's rule, for instance, to a plane figure implies certain assumptions as to the nature of the bounding curve.

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  • 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.

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  • 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"',.

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  • 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.

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  • We can represent waves of longitudinal displacement by a curve, and this enables us to draw very important conclusions in a very simple way.

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  • 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.

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  • 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.

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

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  • 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.

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  • 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.

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  • If a wave travels on without alteration the travelling may be represented by pushing on the displacement curve.

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  • 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.

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  • 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.

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  • 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.

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  • If both vibrate, the point describes a curve which appears continuous through the persistence of the retinal impression.

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  • 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).

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  • The form of the curve is evidently as represented in fig.

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  • The maximum height of the curve HM =a is the amplitude.

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  • If we transfer 0 to A, e=o, and the curve may be represented by y=a sin A x.

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  • 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).

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  • 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.

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  • Let it be represented by a displacement curve Ahbkc. Its periodicity implies that after a certain distance the displacement curve exactly repeats itself.

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  • Let ABCD be drawn at such level that the areas above and below it are equal; then ABCD is the axis of the curve.

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  • 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.

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  • 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...

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  • 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.

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  • 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.

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  • 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.

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  • The dotted curve represents the superposition, which simply doubles each ordinate.

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  • 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.

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  • If the plane does not contain the centre, the curve of intersection is a "small circle," and the solid cut off is a "segment."

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  • The curve of the main arch is a parabola.

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  • The lower portion of the figure is the curve of bending moments under the leading load.

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  • 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.

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  • The heavy continuous line gives the last-mentioned curve for the reverse direction of passage of the loads.

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  • 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.

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  • 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.

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  • Now, except for very short bridges and very unequal loads, a parabola can be found which includes the curve of maximum moments.

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  • This parabola is the curve of maximum moments for a travelling load uniform per ft.

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  • Consider any other point F of the curve, fig.

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  • 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.

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  • 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.

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  • It follows exactly the curve of the mainland, and is continued into Panama, under the name of the Cordillera de Chiriqui.

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

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  • 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.

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  • 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.

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  • 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.

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  • Some of these reverted breeds have developed horns of considerable size, although not showing that regularity of curve distinctive of the wild race.

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  • 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.

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  • 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.

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  • 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.

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  • The upper canines especially are of great size, and curve outwards, forwards and upwards.

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  • The curve itself is sometimes termed the " circumference."

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  • Any line through the centre and terminated at both extremities by the curve, e.g.

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  • 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.

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  • 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.

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  • 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.

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  • The waterside streets, however, follow the curve of the beach.

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  • 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.

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  • 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.

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  • 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.

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  • 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.

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  • 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.

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  • 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.

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  • Newton called attention to the fact that a falling body moves in a curve, diverging slightly from the plumb-line vertical.

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  • If all the ice be melted, we pass along the vapour pressure curve of water OA.

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  • 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.

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  • 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.

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  • 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.

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  • 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.

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  • Experiments on the relation between temperature and concentration are illustrated by projecting the curve OA of fig.

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  • 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.

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  • When this process is complete the temperature rises, and we pass along a new curve giving the equilibrium between anhydrous crystals, solution and vapour.

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  • 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.

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  • Proceeding along the curve in either direction, we come to a non-variant or eutectic point.

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  • 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.

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  • The experi mental curve of solubility is shown in fig.

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  • 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.

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  • When the curve BC is reached, Fe 2 C1 6 - 12H 2 0 separates out, and the solution solidifies.

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  • 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.

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  • 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.

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  • 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.

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  • 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.

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  • All available evidence, from the freezing point curve and from other sources must be scrutinized before an opinion is pronounced.

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  • 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.

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  • 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.

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  • 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.

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  • 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.

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  • Even this correction is not sufficient in solution of sugar, where the theoretical curve II lies below the experimental observations.

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  • 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.

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  • 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.

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  • 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.

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  • 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.

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  • 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.

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  • The area of the loop, which equals the area between the curve and its asymptote, is 3a/2.

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  • 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.

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  • 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.

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  • 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.

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  • 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.

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  • The coast-line extends in a double curve from south-west to northeast, and is formed by a row of sand dunes, 171 m.

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  • 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.

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  • 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.

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  • 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.'

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  • 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.

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  • 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.

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  • 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.

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  • 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.

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  • 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.

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  • 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.

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  • Between the Shetlands and Iceland we cross the curve of maximum frequency, and farther north the frequency diminishes.

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  • 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.

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  • The curve of the arch turns in slightly below the springing, giving a horse-shoe shape.

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  • 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.

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  • She invented and discussed the curve known as the "witch of Agnesi" (q.v.) or versiera.

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  • 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.

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  • 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.

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  • The curve, and the observations it represents, are drawn in fig.

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  • The form of the curve is shown in fig.

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  • Then EG produced meets FP in a point on the curve.

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  • The cartesian parabola is a cubic curve which is also known as the trident of Newton on account of its three-pronged form.

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  • 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.

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  • 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.

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  • 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.

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  • Large flowing leaves of acanthus and other plants were beaten out with wonderful spirit and beauty of curve.

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  • 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.

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  • 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.

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  • The Specific Heat Itself Can Be Deduced Only By Differentiating The Curve Of Observation, Which Greatly Increases The Uncertainty.

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  • 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.

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  • 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.

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  • 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.

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  • 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.

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  • 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.

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  • 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.

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  • 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.

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  • 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.

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  • 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.

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  • We will suppose in the first instance that the curve is plane.

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  • 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.

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  • Next suppose that the curve is rough; and let Fas be the tangential force of friction on s.

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  • Again, take the case of a string under gravity, in contact with a smooth curve in a vertical plane.

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  • This is the intrinsic equation of the curve.

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  • It follows that the osculating plane of the curve formed by the string must contain the normal to the surface, i.

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  • 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.

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  • This is called the curve of positions or space-time curve; its gradient represents the velocity.

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

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  • The curve of positions corresponding to (4) is a parabola, and that of velocities is a straight line.

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  • 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.

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  • 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.

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  • In the case of the pendulum the tension of the string takes the place of the pressure of the curve.

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  • 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.

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  • 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.

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  • 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.

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  • A curve tangential to all the sides of the polygon is the line of pressures.

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  • The path of P is a curve of the kind called epitrochoids.

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  • 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.

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  • 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.

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  • 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.

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  • 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).

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  • This equation gives a relation between the inclination of the curve to the horizon and the height above the level of the liquid.

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  • 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.

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  • Drawings of the different forms of the curve may be found in Thomson and Tait's Natural Philosophy, vol.

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  • 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.

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  • 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.

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  • Hence if N is the point of contact, NP must be normal to the traced curve.

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  • 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.

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  • 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.

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  • 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.

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  • 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.

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  • Beyond them a road branches to the right, sweeping round in a broad curve to the space in front of the temple of Apollo.

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  • 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.

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  • 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.

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  • 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.

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  • 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.

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  • 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.

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  • 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.

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  • 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.

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  • 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.

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  • 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.

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  • 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.

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  • 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.

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  • 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.

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  • 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.

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  • 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.

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  • 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.

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  • 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.

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  • John Wallis discussed both the history and properties of the curve in a tract De cycloide published at Oxford in 1659.

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  • Honore Fabri (Synopsis geometrica, 1669) treated of the curve and enumerated many theorems concerning it.

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  • 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.

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  • 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.

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  • 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.

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  • The mechanical properties of the cycloid were investigated by Christiaan Huygens, who proved the curve to be tautochronous.

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  • 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.

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  • The curve is shown in fig.

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  • 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.

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  • 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.

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  • 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.

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  • 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.

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  • 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.

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  • 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.

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  • 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.

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  • 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.

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  • 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.

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  • 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.

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  • 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.

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  • Another Asiatic species is the great shou (C. affinis) of the Chumbi Valley, in which the antlers curve forwards in a remarkable manner.

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  • 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.

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  • 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.

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  • 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.

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  • 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.

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  • 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.

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  • 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.

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  • 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.

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  • The most elevated portion of the innermost range, the Drakensberg follows the curve of the coast from south to north-east.

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  • 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.

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  • 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.

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  • 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.

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  • They are directed outwards, and curve in an open spiral, with the tips directed outwards.

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  • 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.

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  • 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.

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  • 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.

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  • 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.

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  • He announced that the curve required in the first problem must be a cycloid, and he gave a method of determining it.

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  • The curve consists of one branch entirely to the left of the line x= 2a and having the axis of y as an asymptote.

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  • 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."

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  • A curve is a line, or continuous singly infinite system of points.

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  • We consider in the first instance, and chiefly, a plane curve described according to a law.

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  • 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.

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  • 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.

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  • 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.

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  • Every curve thus arbitrarily defined has its own properties; and there was not any principle of classification.

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  • 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.

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  • Observe that the distinctive feature is in the exclusive use of such determination of a curve by means of its equation.

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  • 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.

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  • We obtain from the equation the notion of an algebraical as opposed to a transcendental curve, viz.

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  • The equation is sometimes given, and may conveniently be used, in an irrational form, but we always imagine it reduced to the foregoing rational and integral form, and regard this as the equation of the curve.

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  • And we have hence the notion of a curve of a given order, viz.

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  • It is to be noticed here that the axes of co-ordinates may be any two lines at right angles to each other whatever; and that the equation of a curve will be different according to the selection of the axes of co-ordinates; but the order is independent of the axes, and has a determinate value for any given curve.

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  • A curve of the first order is a right line; and conversely every right line is a curve of the first order.

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  • A curve of the second order is a conic, and is also called a quadric curve; and conversely every conic is a curve of the second order or quadric curve.

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  • A curve of the third order is called a cubic; one of the fourth order a quartic; and so on.

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  • A curve of the order na has for its equation (*1 x, y, 1)m=o; and when the coefficients of the function are arbitrary, the curve is said to be the general curve of the order in.

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  • Each of these last equations represents a curve of the first order, or right line; and the original equation represents this pair of lines, viz.

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  • 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.

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  • 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.

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  • Reverting to the purely plane theory, infinity is a line, related like any other right line to the curve, and thus intersecting it in m points, real or imaginary, distinct or coincident.

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  • We have in it a particular kind of correspondence of two points on a cubic curve, viz.

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  • It may be remarked that in Poncelet's memoir on reciprocal polars, above referred to, we have the theorem that the number of tangents from a point to a curve of the order m, or say the class of the curve, is in general and at most = m(m - 1), and that he mentions that this number is subject to reduction when the curve has double points or cusps.

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  • 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.

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  • Plucker first gave a scientific dual definition of a curve, viz.; " A curve is a locus generated by a point, and enveloped by a line - the point moving continuously along the line, while the line rotates continuously about the point "; the point is a point (ineunt.) of the curve, the line is a tangent of the curve.

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  • And, assuming the above theory of geometrical imaginaries, a curve such that m of its points are situate in an arbitrary line is said to be of the order m; a curve such that n of its tangents pass through an arbitrary point is said to be of the class n; as already appearing, this notion of the order and class of a curve is, however, due to Gergonne.

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  • Thus the line is a curve of the order i and class o; and corresponding dually thereto, we have the point as a curve of the order o and class 1.

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  • But as regards the representation of a curve by an equation, the case is very different.

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  • It is implied in Pliicker's theorem that, m, n, signifying as above in regard to any curve, then in regard to the reciprocal curve, n, m, will have the same significations, viz.

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  • The expression 2 is that of the number of the disposable constants in a curve of the order m with nodes and cusps (in fact that there shall be a node is I condition, a cusp 2 conditions) and the equation (9) thus expresses that the curve and its reciprocal contain each of them the same number of disposable constants.

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  • For a curve of the order the expression Zm(m - I) - 6 - K is termed the " deficiency " (as to this more hereafter); the equation (to) expresses therefore that the curve and its reciprocal have each of them the same deficiency.

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  • With regard to the demonstration of Pliicker's equations it is to be remarked that we are not able to write down the equation in point-co-ordinates of a curve of the order m, having the given numbers 6 and of nodes and cusps.

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  • We can only use the general equation (*fix, y, z) m = o, say for shortness u= o, of a curve of the mth order, which equation, so long as the coefficients remain arbitrary, represents a curve without nodes or cusps.

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  • Seeking then, for this curve, the values, n, e, of the class, number of inflections, and number of double tangents, - first, as regards the class, this is equal to the number of tangents which can be drawn to the curve from an arbitrary point, or what is the same thing, it is equal to the number of the points of contact of these tangents.

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  • The points of contact are found as the intersections of the curve u= o by a curve depending on the position of the arbitrary point, and called the " first polar " of this point; the order of the first polar is = m - r, and the number of intersections is thus =m(m - I).

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  • But it can be shown, analytically or geometrically, that if the given curve has a node, the first polar passes through this node, which therefore counts as two intersections, and that if the curve has a cusp, the first polar passes through the cusp, touching the curve there, and hence the cusp counts as three intersections.

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  • But, as is evident, the node or cusp is not a point of contact of a proper tangent from the arbitrary point; we have, therefore, for a node a diminution and for a cusp a diminution 3, in the number of the intersections; and thus, for a curve with 6 nodes and K cusps, there is a diminution 26+3K, and the value of n is n= m (m - I)-26-3K.

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  • Secondly, as to the inflections, the process is a similar one; it can be shown that the inflections are the intersections of the curve by a derivative curve called (after Ludwig Otto Hesse who first considered it) the Hessian, defined geometrically as the locus of a point such that its conic polar (§ 8 below) in regard to the curve breaks up into a pair of lines, and which has an equation H = o, where H is the determinant formed with the second differential coefficients of u in regard to the variables (x, y, z); H= o is thus a curve of the order 3 (m - 2), and the number of inflections is =3m(m-2).

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  • But if the given curve has a node, then not only the Hessian passes through the node, but it has there a node the two branches at which touch respectively the two branches of the curve; and the node thus counts as six intersections; so if the curve has a cusp, then the Hessian not only passes through the cusp, but it has there a cusp through which it again passes, that is, there is a cuspidal branch touching the cuspidal branch of the curve, and besides a simple branch passing through the cusp, and hence the cusp counts as eight intersections.

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  • The node or cusp is not an inflection, and we have thus for a node a diminution 6, and for a cusp a diminution 8, in the number of the intersections; hence for a curve with 6 nodes and cusps, the diminution is = 66+8K, and the number of inflections is c= 3m(m - 2) - 66 - 8K.

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  • Thirdly, for the double tangents; the points of contact of these are obtained as the intersections of the curve by a curve II = o, which has not as yet been geometrically defined, but which is found analytically to be of the order (m-2) (m 2 -9); the number of intersections is thus = m(rn - 2) (m 2 - 9); but if the given curve has a node then there is a diminution =4(m2 - m-6), and if it has a cusp then there is a diminution =6(m2 - m-6), where, however, it is to be noticed that the factor (m2 - m-6) is in the case of a curve having only a node or only a cusp the number of the tangents which can be drawn from the node or cusp to the curve, and is used as denoting the number of these tangents, and ceases to be the correct expression if the number of nodes and cusps is greater than unity.

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  • To complete Pliicker's theory it is necessary to take account of compound singularities; it might be possible, but it is at any rate difficult, to effect this by considering the curve as in course of description by the point moving along the rotating line; and it seems easier to consider the compound singularity as arising from the variation of an actually described curve with ordinary singularities.

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  • So that, in fact, Pliicker's equations properly understood apply to a curve with any singularities whatever.

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  • By means of Pliicker's equations we may form a table - The table is arranged according to the value of in; and we have m=o, n= r, the point; m =1, n =o, the line; m=2, n=2, the conic; of m = 3, the cubic, there are three cases, the class being 6, 4 or 3, according as the curve is without singularities, or as it has 1 node or r cusp; and so of m =4, the quartic, there are ten cases, where observe that in two of them the class is = 6, - the reduction of class arising from two cusps or else from three nodes.

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  • The whole theory of the inflections of a cubic curve is discussed in a very interesting manner by means of the canonical form of the equation x +y +z +6lxyz= o; and in particular a proof is given of Plucker's theorem that the nine points of inflection of a cubic curve lie by threes in twelve lines.

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  • It may be noticed that the nine inflections of a cubic curve represented by an equation with real coefficients are three real, six imaginary; the three real inflections lie in a line, as was known to Newton and Maclaurin.

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  • The expression 2m(m - 2) (m - 9) for the number of double tangents of a curve of the order in was obtained by Plucker only as a consequence of his first, second, fourth and fifth equations.

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  • Mag., 1858; considering the m - 2 points in which any tangent to the curve again meets the curve, he showed how to form the equation of a curve of the order (m - 2), giving by its intersection with the tangent the points in question; making the tangent touch this curve of the order (m - 2), it will be a double tangent of the original curve.

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  • 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.

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  • A quartic curve has 28 double tangents, their points of contact determined as the intersections of the curve by a curve II = o of the order 14, the equation of which in a very elegant form was first obtained by Hesse (1849).

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  • The Hessian A has just been spoken of as a covariant of the form u; the notion of invariants and covariants belongs rather to the form u than to the curve u=o represented by means of this form; and the theory may be very briefly referred to.

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  • A curve u=o may have some invariantive property, viz.

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  • Similarly, if we have a curve U= o derived from the curve u = o in a manner independent of the particular axes of co-ordinates, then from the transformed equation u' = o deriving in like manner the curve U' = o, the two equations U= o, U' = o must each of them imply the other; and when this is so, U will be a covariant of u.

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  • The theory of the invariants and covariants of a ternary cubic function u has been studied in detail, and brought into connexion with the cubic curve u = o; but the theory of the invariants and covariants for the next succeeding case, the ternary quartic function, is still very incomplete.

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  • In further illustration of the Pluekerian dual generation of a curve, we may consider the question of the envelope of a variable curve.

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  • 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.

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  • To develop the theory, consider the curve corresponding to any particular value of the parameter; this has with the consecutive curve (or curve belonging to the consecutive value of the parameter) a certain number of intersections and of common tangents, which may be considered as the tangents at the intersections; and the so-called envelope is the curve which is at the same time generated by the points of intersection and enveloped by the common tangents; we have thus a dual generation.

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  • 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.

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  • 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.

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  • 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.

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

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  • Any one of these is a " parallel " of the given curve; and it can be obtained as the envelope of a circle of constant radius having its centre on the given curve.

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  • We have in like manner, as derivatives of a given curve, the caustic, catacaustic or diacaustic as the case may be, and the secondary caustic, or curve cutting at right angles the reflected or refracted rays.

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  • For real figures we have the general theorem that imaginary intersections, &c., present themselves in conjugate pairs; hence, in particular, that a curve of an even order is met by a line in an even number (which may be = o) of points; a curve of an odd order in an odd number of points, hence in one point at least; it will be seen further on that the theorem may be generalized in a remarkable manner.

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  • Again, when there is in question only one pair of points or lines, these, if coincident, must be real; thus, b line meets a cubic curve in three points, one of them real, and other two real or imaginary; but if two of the intersections coincide they must be real, and we have a line cutting a cubic in one real point and touching it in another real point.

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  • 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.

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  • 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.

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  • A form which presents itself is when two ovals, one inside the other, unite, so as to give rise to a crunode - in default of a better name this may be called, after the curve of that name, a limacon.

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  • Names may also be used for the different forms of infinite branches, but we have first to consider the distinction of hyperbolic and parabolic. The leg of an infinite branch may have at the extremity a tangent; this is an asymptote of the curve, and the leg is then hyperbolic; or the leg may tend to a fixed direction, but so that the tangent goes further and further off to infinity, and the leg is then parabolic; a branch may thus be hyperbolic or parabolic as to its two legs; or it may be hyperbolic as to one leg and parabolic as to the other.

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  • And we thus see that the two hyperbolic legs belong to a simple intersection of the curve by the line infinity.

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  • And we thus see that the two parabolic legs represent a contact of the line infinity with the curve, - the point of contact being of course the point at infinity determined by the common direction of the two legs.

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  • In general a cone consists of one or more single or twin-pair sheets, and if we consider the section of the cone by a plane, the curve consists of one or more complete branches, or say circuits, each of them the section of one sheet of the cone; thus, a cone of the second order is one twin-pair sheet, and any section of it is one circuit composed, it may be, of two branches.

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  • But although we thus arrive by projection at the notion of a circuit, it is not necessary to go out of the plane, and we may (with Zeuthen, using the shorter term circuit for his complete branch) define a circuit as any portion (of a curve) capable of description by the continuous motion of a point, it being understood that a passage through infinity is permitted.

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  • And we then say that a curve consists of one or more circuits; thus the right line, or curve of the first order, consists of one circuit; a curve of the second order consists of one circuit; a cubic curve consists of one circuit or else of two circuits.

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  • It may be added that there are on the odd circuit three inflections, but on the even circuit no inflection; it hence also appears that from any point of the odd circuit there can be drawn to the odd circuit two tangents, and to the even circuit (if any) two tangents, but that from a point of the even circuit there cannot be drawn (either to the odd or the even circuit) any real tangent; consequently, in a simplex curve the number of tangents from any point is two; but in a complex curve the number is four, or none, - f our if the point is on the odd circuit, none if it is on the even circuit.

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  • It at once appears from inspection of the figure of a non-singular cubic curve, which is the odd and which the even circuit.

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  • The singular kinds arise as before; in the crunodal and the cuspidal kinds the whole curve is an odd circuit, but in an acnodal kind the acnode must be regarded as an even circuit.

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  • We may consider in relation to a curve, not only the line infinity, but also the circular points at infinity; assuming the curve to be real, these present themselves always conjointly; thus a circle is a conic passing through the two circular points, and is thereby distinguished from other conics.

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  • We may from each of the circular points draw tangents to a given curve; the intersection of two such tangents (belonging of course to the two circular points respectively) is a focus.

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  • There are in some cases points termed centres, or singular or multiple foci (the nomenclature is unsettled), which are the intersections of improper tangents from the two circular points respectively; thus, in the circular cubic, the tangents to the curve at the two circular points respectively (or two imaginary asymptotes of the curve) meet in a centre.

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  • 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.

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  • When D = o, the curve is said to be unicursal, when = i, bicursal, and so on.

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  • In particular if D =o, that is, if the given curve be unicursal, the transformed curve is a line, 4 is a mere linear function of 0, and the theorem is that the co-ordinates x, y, z of a point of the unicursal curve can be expressed as proportional to rational and integral functions of 0; it is easy to see that for a given curve of the order m, these functions of 0 must be of the same order m.

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  • And so if D =2, then the transformed curve is a nodal quartic; 4 can be expressed as the square root of a sextic function of 0 and the theorem is, that the co-ordinates x, y, z of a point of the tricursal curve can be expressed as proportional to rational and integral functions of 0, and of the square root of a sextic function of 0.

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  • It is a form of the theorem for the case D = r, that the coordinates x, y, z of a point of the bicursal curve, or in particular the co-ordinates of a point of the cubic, can be expressed as proportional to rational and integral functions of the elliptic functions snu, cnu, dnu; in fact, taking the radical to be r -0 2 .r - k 2 0 2, and writing 8 =snu, the radical becomes = cnu, dnu; and we have expressions of the form in question.

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  • 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.

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  • Imagine a curve of order m, deficiency D, and let the corresponding points P, P' be such that the line joining them passes through a given point 0; this is an (m - m-1) correspondence, and the value of k is=1, hence the number of united points is =2m-2+2D; the united points are the points of contact of the tangents from 0 and (as special solutions) the cusps, and we have thus the relation or, writing D=2(m - i)(m-2) - S - K, this is n=m(m - i)-23-3K, which is right.

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  • We have thus a means of geometrical representation for the portions, as well imaginary as real, of any real or imaginary curve.

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  • Supposing 0 indefinitely small, we have what may be called the penultimate curve, and when 0=o the ultimate curve.

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  • 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.

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  • The ultimate curve together with its summits may be regarded as a degenerate form of the curve u=o.

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  • Observe that the positions of the summits depend on the penultimate curve u=o, viz.

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  • Taking P 1 =o to have 3 1 nodes and K 1 cusps, and therefore its class n 1 to be=m 1 2 - m 1 25 1 -3K,, the expression for the number of tangents to the penultimate curve is = (a1 2 - a,) m1 2 + (a2 2 - a2)m2 2 +

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  • Such a curve may be considered as described by a point, moving in a line which at the same time rotates about the point in a plane which at the same time rotates about the line; the point is a point, the line a tangent, and the plane an osculating plane, of the curve; moreover the line is a generating line, and the plane a tangent plane, of a developable surface or torse, having the curve for its edge of regression.

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  • Analogous to the order and class of a plane curve we have the order, rank and class of the system (assumed to be a geometrical one), viz.

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  • The system has singularities, and there exist between m, r, is and the numbers of the several singularities equations analogous to Pliicker's equations for a plane curve.

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  • 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.

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  • The Ligurian Apennines extend as far as the pass of La Cisa in the upper valley of the Magra (anc. Macra) above Spezia; at first they follow the curve of the Gulf of Genoa, and then run east-south-east parallel to the coast.

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  • Among these subjects were the transit of Mercury, the Aurora Borealis, the figure of the earth, the observation of the fixed stars, the inequalities in terrestrial gravitation, the application of mathematics to the theory of the telescope, the limits of certainty in astronomical observations, the solid of greatest attraction, the cycloid, the logistic curve, the theory of comets, the tides, the law of continuity, the double refraction micrometer, various problems of spherical trigonometry, &c. In 1742 he was consulted, with other men of science, by the pope, Benedict XIV., as to the best means of securing the stability of the dome of St Peter's, Rome, in which a crack had been discovered.

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  • Humped cattle are widely spread over Africa, Madagascar and India, and form a distinct species, Bos indicus, characterized by the presence of a fleshy hump on the shoulders, the convexity (instead of concavity) of the first part of the curve of the horns, the very large size of the dewlap, and the general presence of white rings round the fetlocks, and light circles surrounding the eyes.

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  • If projected with this velocity in any direction the point of projection will be at the end of the minor axis of the orbit, because this is the only point of an ellipse of which the distance from the focus is equal to the semi-major axis of the curve, and therefore the only point at which the distance of the body from the sun is equal to its mean distance.

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  • Actually the curve in which it moves is nearly a circle; but the distance varies slightly owing to the minute secular variation in the position of the ecliptic, caused by the action of the planets.

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  • It was further investigated by John Wallis, Christiaan Huygens (who determined the length of any arc in 1657), and Pierre de Fermat (who evaluated the area between the curve and its asymptote in 1661).

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  • The curve is symmetrical about the axis of x, and consists of two infinite branches asymptotic to the line BT and forming a cusp at the origin.

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  • The area between the curve and its asymptote is 37ra 2, i.e.

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  • West of the Urumchi gap, the Bogdo-ola is continued in the double range of the Iren-khabirga Mountains (11,500 ft.), which curve to the north-west and finally, under the name of the Talki Mountains, merge into the Boro-khoro range.

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  • The orbit of the moon around the earth, though not a fixed curve of any class, is elliptical in form, and may be represented by an ellipse which is constantly changing its form and position, and has the earth in one of its foci.

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  • The plateau of the Istrian Karst is prolonged in several of the bare and desolate mountain chains between the Save and the Adriatic, notably the Great and Little Kapella (or Kapela), which link together the Karst and the Dinaric Alps, culminating in Biela Lazica (5029 ft.); the Pljesevica or Plisevica Planina (5410 ft.), overlooking the valley of the river Una; and the Velebit Planina, which follows the westward curve of the coast, and rises above the sea in an abrupt wall, unbroken by any considerable bay or inlet.

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  • Skeat takes the ultimate root to be kar, to move, especially in a circular motion, seen in "curve," "circle," &c. The word "worm" is applied to many objects resembling the animals in having a spiral shape or motion, as the spiral thread of a screw, or the spiral pipe through which vapour is passed in distillation.

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  • Traversing this in a majestic northward curve and receiving vast supplies of water from many great tributaries, it finally turns south-west and cuts a way to the Atlantic Ocean through the western highlands.

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  • Among the most fashionable streets are Mount Curve, Clifton and Park avenues, all in the " West Division "or south-western quarter of the city.

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  • It admits of several definitions framed according to the aspect from which the curve is considered.

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  • The relation of the ellipse to the other conic sections is treated in the articles Conic Section and Geometry; in this article a summary of the properties of the curve will be given.

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  • If SX be divided at A so that SA/AX = e, then A is a point on the curve.

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  • It is obvious that the curve is symmetrical about AA'.

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  • If AA' be bisected at C, and the line BCB' be drawn perpendicular to AA', then it is readily seen that the curve is symmetrical about this line also; since if we take S' on AA' so that S'A' =SA, and a.

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  • If B and B' be points on the curve, BB' is the minor axis and C the centre of the curve.

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  • Metrical relations between the axes, eccentricity, distance between the foci, and between these quantities and the co-ordinates of points on the curve (referred to the axes and the centre), and focal distances are readily obtained by the methods of geometrical conics or analytically.

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  • From any point without the curve two, and only two, tangents can be drawn; if OP, OP' be two tangents from 0, and S, S' the foci, then the angles OSP, OSP' are equal and also SOP, S'OP'.

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  • The middle points of a system of parallel chords is a straight line, and the tangent at the point where this line meets the curve is parallel to the chords.

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  • The thread is now stretched taut by a pencil, and the pencil moved; the curve traced out is the desired ellipse.

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  • By varying the position of R other points can be found, and, since the curve is symmetrical about both the major and minor axes, it is obvious that any point may be reflected in both the axes, thus giving 3 additional points.

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  • Then the intersection of EB and DB' determines a point P on the (true) curve.

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  • An arc with centre 0 and radius OB forms part of a curve.

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  • On the north-east and east, where the edge of the table-land sweeps round in a wide curve, the surface sinks in broad terraces to the valley of the Ebro and the Bay of Valencia, and is crowned by more or less isolated mountains, some of which have been already mentioned.

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  • The stomach at its point of junction with the rectum presents an S-shaped ventro-dorsal curve.

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  • It describes a wide curve eastwards past Soria, then flows westward across the Castilian table-land, passing south of Valladolid, with Toro and Zamora on its right bank; then from a point 3 m.

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  • During his tenure of this chair he published two volumes of a Course of Mathematics - the first, entitled Elements of Geometry, Geometrical Analysis and Plane Trigonometry, in 1809, and the second, Geometry of Curve Lines, in 1813; the third volume, on Descriptive Geometry and the Theory of Solids was never completed.

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  • This ratio, known as the eccentricity, determines the nature of the curve; if it be greater than unity, the conic is a hyperbola; if equal to unity, a parabola; and if less than unity, an ellipse.

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  • The points in which the cutting plane intersects the sides of the triangle are the vertices of the curve; and the line joining these points is a diameter which Apollonius named the latus transversum.

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  • Then the square of the ordinate intercepted between the diameter and the curve is equal to the rectangle contained by the portion of the diameter between the first vertex and the foot of the ordinate, and the segment of the ordinate intercepted between the diameter and the line joining the extremity of the latus rectum to the second vertex.

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  • This is shown by the curve AB, and is the useless work represented by the expression (q 2 -q 1)/r.

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  • If the contact springs can be moved round the disk so as to vary the instant of contact, we can plot out the value of the observed instantaneous voltage of the machine or circuit in a wavy curve, showing the wave form of the electromotive force of the alternator.

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  • As the brushes are slowly shifted over on the revolving contact so as to select different phases of the alternating electromotive force, the pen follows and draws a curve delineating the wave form of that electromotive force or current.

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  • The oscillograph can be made to exhibit optically the form of the current curve in non-cyclical phenomena, such as the discharge of a condenser.

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  • The scenery became more spectacular with each rounded curve.

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  • As Dean rounded a curve, he caught sight of the tail end of a white vehicle speeding down the cliff-hanging road on the far side of the deep valley—a sheriff's white Blazer was his first impression.

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  • Before his mind had a chance to act, two yellow beams of a headlight broke around the curve of the roadway a half-mile below him and began a slow climb to where he stood.

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  • It was not possible to define a clear response curve of PD excretion to purine absorption from that data set.

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  • Yet although the curve of burned lands parallels the curve of afforested lands, afforested lands, afforestation is only a minor part of the problem.

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  • She also tends to curve to the right and has patchy alopecia.

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  • Along the various layers of rock a curve down is called a syncline, a curve in an upwards is called an anticline.

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  • The height of each curve is the semi-major axis of that planet, a measure of its distance from the star.

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  • The sequence of diagrams above shows the stable set, denoted by the blue curve, undergoing a type-2 contact bifurcation.

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  • The UK has a sign warning of " adverse camber " on a curve.

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  • The rotation curve is a plot of the orbital velocity of the clouds around the galactic center vs. their distance from the Galaxy center.

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  • Negative inotropes like acetyl choline will move the curve to the right.

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  • Also plotted is a curve for the minimum buckling stress coefficient.

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  • As the track begins to curve to the left a small path enters a small copse to the left.

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  • Since it opened in 2004, Curve has joined the small coterie of restaurants in Canary Wharf that offers serious wining & dining.

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  • Some more recent subjects (for example AES and elliptic curve cryptography) are not well covered here.

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  • Thus there will be an upward sloping demand curve.

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  • This climbing up the PES leads to the decreased slope of the caloric curve at higher energies.

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  • It fits snugly into your lumbar curve, relieving strain on the lower back muscles.

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  • The program includes the non-linear elastic behavior beyond the limit of proportionality using the generalized stress-strain curve of ESDU 76016.

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  • Note that multiple plots on the calibration curve always use the default curve.

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  • The graph below shows the ' normal ' shape of a yield curve.

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  • These points of departure approaches mad better use of all the data to estimate the dose-response curve.

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  • Each titration curve should be given a unique label.

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  • They said a wheel broke on a curve as the train approached the Jesus underground station and two train carriages derailed in the tunnel.

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  • It gives a continuous curve which is of infinite length and nowhere differentiable.

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  • Stone Arthur is out of sight below the curve of the summit dome.

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  • The design of the phantom assures both films and alanine dosimeters are accurately located at the same point on the depth dose curve.

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  • Churchyard The churchyard is now a medium-sized, largely rectilinear enclosure, with a visible curve only on the east.

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  • Rocking curve FWHM indicate a resolution of less than 1 eV at 2450 eV and less than 3 eV at 4000 eV photon energy.

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  • The bell-shaped curve forms a " spectral envelope " .

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  • If there is a change in inflation expectations in the economy we see a shift in the Phillips Curve.

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  • Trustees are asked to consider these recommendations for the Learning Curve strategic framework.

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  • The Dynamic Avon The single cable forms a graceful curve to support bridge girders without blocking the view creating a gorgeous landscape.

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  • This is because the gradient of a curve at a point is equal to the gradient of the tangent at that point.

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  • Craft Wearable pieces of art including a headpiece that is molded to the natural curve of the head.

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  • While this design may have been traditional, the shape of the new 50p coin, an equilateral curve heptagon, was revolutionary.

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  • The curve will be exactly the same as when you add hydrochloric acid to sodium hydroxide.

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  • A slight imperfection is visible where the curve reverses under the word " Victoria " .

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  • Summary An indifference curve is a locus of points about which the individual feels indifferent.

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  • The marginal cost curve must also intersect the average cost at its minimum.

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  • They should also be narrow and curve inwards, being very supple with a fine and velvety texture.

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  • It bent around in such a curve as to end in a wide angle toward two degrees forty minutes north latitude.

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  • Mathematica is intimidating at first; it has a steep learning curve.

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