# Curve sentence example

curve

- His finger left her jaw and softly followed the curve of her neck.
- The sharp curve arrived before her memory of it.
- The black dress she wore fit her like a second skin, outlining every curve, dip and nook of her body.
- This curve with the values reduced from metres to feet is reproduced below.
- His lips followed, softly pressing against the curve of her neck.Advertisement
- It was still light enough to see across the gorge when an opening in the trees allowed, but the long swing to the far end of valley was away from the direction the vehicle had driven and blocked from sight by the curve of the canyon.
- The first curve frightened the hell out of him and he knew the brake pressure necessary to slow him from this speed could not be engaged all the way down the mountain without overheating the tiny pads to the point of ineffectiveness.
- He began first a short ascent, then a drop to a sharp curve he nearly missed, causing him to reduce his speed further.
- The amount of superelevation required to prevent derailment at a curve can be calculated under perfect running conditions, given the radius of curvature, the weight of the vehicle, the height of the centre of gravity, the distance between the rails, and the speed; but great experience 1 See The Times Engineering Supplement (August 22, 1906), p. 265.
- Dean pulled out of the curve, searching ahead for a glimpse of his quarry as he continued to hug the right side of the narrow roadway.Advertisement
- 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.
- As she rounded the curve in the staircase, the room became silent.
- It flashed, silver glinting off its graceful curve.
- He grabbed the door handle as she spun around a curve.
- She rounded a curve and slammed on the breaks.Advertisement
- This three-peaked curve is not wholly pecuiiar to Paris, being seen, for instance, at Lisbon in summer.
- His reputation mainly rests on his Introduzione ad una teoria geometrica delle curve piane, which proclaims him as a follower of the Steinerian or synthetical school of geometricians.
- If the asymptotes be perpendicular, or, in other words, the principal axes be equal, the curve is called the rectangular hyperbola.
- An important property is: the difference of the focal distances of any point on the curve equals the transverse axis.
- 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.Advertisement
- When a train is running round a curve the centrifugal force which comes into play tends to make its wheel-flanges press against the outer rail, or even to capsize it.
- The celluloid sheet is laid upon the squared paper, and in plotting a curve horizontal distances are reckoned from the proper demagnetization line instead of from the vertical axis.
- The parallel ranges extend east and west with a slight southerly curve towards their centres.
- Go up the hill and watch for cars so you can warn anyone before they get to the curve.
- 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.Advertisement
- 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.
- 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.
- The plotting of the torque curve is laborious, but the average torque acting, which is all that is required for the purposes of this article, can be found quite simply, thus: - Let p be the mean effective pressure acting in one cylinder, a, the area of the cylinder, and 1, the stroke.
- Another of Roberval's discoveries was a very general method of drawing tangents, by considering a curve as described by a moving point whose motion is the resultant of several simpler motions.
- It flows east and south in a wide curve, through a broad upper valley past Chippenham and Melksham, after which it turns abruptly west to Bradford-on-Avon, receives the waters of the Frome from the south, and enters the beautiful narrow valley in which lie Bath and Bristol.Advertisement
- It is also the inverse of the same curve for the same point.
- The name lemniscate is sometimes given to any crunodal quartic curve having only one real finite branch which is symmetric about the axis.
- If a be greater than b the curve resembles fig.
- 2 and is sometimes termed the fishtail-lemniscate; if a be less than b, the curve resembles fig.
- The centre is a conjugate point (or acnode) and the curve resembles fig.Advertisement
- In this case the centre is a crunode and the curve resembles fig.
- The instrument can be provided with a curve or table showing the current corresponding to each angular displacement of the torsion head.
- This paper is principally based on the following general theorem, which is a remarkable extension of Pascal's hexagram: "If a polygon move so that each of its sides passes through a fixed point, and if all its summits except one describe curves of the degrees m, n, p, &c., respectively, then the free summit moves on a curve of the degree 2mnp. ..
- If the attraction of a central body is not the only force acting on the moving body, the orbit will deviate from the form of a conic section in a degree depending on the amount of the extraneous force; and the curve described may not be a re-entering curve at all, but one winding around so as to form an indefinite succession of spires.
- Let the curve represent an elliptic orbit, AB being the major axis, DE the minor axis, and F the focus in which the centre of attraction is situated, which centre we shall call the sun.Advertisement
- The polar equation is r= I -f - 2 cos 0 and the form of the curve is shown in the figure.
- The overheating curve of rhombic sulphur extends along the curve AC, where C is the melting-point of monoclinic sulphur.
- From B the curve of equilibrium (BD) between rhombic and liquid sulphur proceeds; and from C (along CE) the curve of equilibrium between liquid sulphur and sulphur vapour.
- Of especial interest is the 0 curve BD; along this line liquid and rhombic sulphur are in equilibrium, which means that at above 131° and 400 atmospheres the rhombic (and not the monoclinic) variety would separate from liquid sulphur.
- Two cases then arise: (I) the properties may be expressed as linear functions of the composition, the terminal values being identical with those obtained for the individual components, and there being a break in the curve corresponding to the absence of mixed crystals; or (2) similar to (I) except that different values must be assigned to the terminal values in order to preserve collinearity.Advertisement
- For the subjects under this heading see the articles CONIC SECTIONS; CIRCLE; CURVE; GEOMETRICAL CONTINUITY; GEOMETRY, Axioms of; GEOMETRY, Euclidean; GEOMETRY, Projective; GEOMETRY, Analytical; GEOMETRY, Line; KNOTS, MATHEMATICAL THEORY OF; MENSURATION; MODELS; PROJECTION; Surface; Trigonometry.
- The surface of the inland ice forms in a transverse section from the west to the east coast an extremely regular curve, almost approaching an arc of a wide circle, which along Nansen's route has its highest ridge somewhat nearer the east than the west coast.
- The freezing point curve usually lies below the electrical one, but approaches it as dilution is increased.2 Returning once more to the consideration of the first relation, which deals with the comparison between the number of ions and the number of pressure-producing particles in dilute solution, one caution is necessary.
- The direction of magnetic induction may be indicated by lines of induction; a line of induction is always a closed curve, though it may possibly extend to and return from infinity.
- Since the induction B is equal to H 47rI, it is easy from the results of experiments such as that just described to deduce the relation between B and H; a curve indicating such relation is called a curve of induction.Advertisement
- 12 shows the nature of the course taken by the curve when the magnetizing current, after having been raised to the value corresponding to the point a, is diminished by steps until it is nothing, and then gradually increased in the reverse direction.
- The downward course of the curve is, owing to hysteresis, strikingly different from its upward course, and when the magnetizing force has been reduced to zero, there is still remaining an induction of 7500 units.
- The closed figure a c d e a is variously called a hysteresis curve or diagram or loop. The area f HdB enclosed by it represents the work done in carrying a cubic centimetre of the iron through the corresponding magnetic cycle; expressed in ergs this work is I HdB.
- When it is desired to obtain a simple curve of induction, such as that in fig.
- When a hysteresis curve is to be obtained, the procedure is as follows: The current is first adjusted by means of R to such a strength as will fit it to produce the greatest + and - values of the magnetizing force which it is intended to apply in the course of the cycle; then it is reversed several times, and when the range of the galvanometer throws has become constant, half the extent of an excursion indicates the induction corresponding to the extreme value of H, and gives the point a in the curve fig.
- The galvanometer throw which results from the change of current measures the amount by which the induction is reduced, and thus a second point on the curve is found.
- In a similar manner, by giving different values to the resistance 4 F R, any desired number of points R= between a and c in the curve can FIG.
- 12, and the determination of the downward limb of the curve is therefore completed.
- As the return curve, shown in fig.
- On the other hand, the form of the third curve, with its large intercepts on the axes of H and B, denotes that the specimen to which it relates possesses both retentiveness and coercive force in a high degree; such a metal would be chosen for making good permanent magnets.
- The curve thus constructed should be a straight line inclined to the horizontal axis at an angle 0, the tangent of which is 1.6.
- But though a formula of this type has no physical significance, and cannot be accepted as an equation to the actual curve of W and B, it is, nevertheless, the case that by making the index e =1.6, and assigning a suitable value to r t, a formula may be obtained giving an approximation to the truth which is sufficiently close for the ordinary purposes of electrical engineers, especially when the limiting value of B is neither very great nor very small.
- These are to be regarded merely as typical specimens, for the details of a curve depend largely upon the physical condition and purity of the material; but they show at a glance how far the several metals differ from and resemble one another as regards their magnetic properties.
- During the first stage, when the magnetizing force is small, the magnetization (or the induction) increases rather slowly with increasing force; this is well shown by the nickel curve in the diagram, but the effect would be no less conspicuous in the iron curve if the abscissae were plotted to a larger scale.
- During the second stage small increments of magnetizing force are attended by relatively large increments of magnetization, as is indicated by the steep ascent of the curve.
- Then the curve bends over, forming what is often called a " knee," and a third stage is entered upon, during which a considerable increase of magnetizing force has little further effect upon the magnetization.
- Rowland, believing that the curve would continue to fall in a straight line meeting the horizontal axis, inferred that the induction corresponding to the point B-about 17,500-was the highest I Phil.
- It has, however, been shown that, if the magnetizing force is carried far enough, the curve always becomes convex to the axis instead of meeting it.
- The curve for cobalt is a very remarkable one.
- The effect produced by a current is exactly opposite to that of tension, raising the elongation curve instead of depressing it.
- Some experiments were next undertaken with the view of ascertaining how far magnetic changes of length in iron were dependent upon the hardness of the metal, and the unexpected result was arrived at that softening produces the same effect as tensile stress; it depresses the elongation curve, diminishing the maximum extension, and reducing the " critical value " of the magnetizing force.
- The experiments were not sufficiently numerous to indicate whether, as is possible, there is a critical degree of hardness for which the height of the elongation curve is a maximum.
- Among other things, it was found that the behaviour of cast cobalt was entirely changed by annealing; the sinuous curve shown in Fig.
- Ewing's independent experiments showed that the magnetization curve for a cobalt rod under a load of 16.2 kilogrammes per square mm.
- As regards the higher temperatures, the chief point of interest is the observation that the curve of magnetization for annealed cobalt shows a small depression at about 450°, the temperature at which they had found the sign of the length-change to be reversed for all fields.
- When the curve after its steep descent has almost reached the axis, it bends aside sharply and becomes a nearly horizontal straight line; the authors suggest that the critical temperature should be defined as that corresponding to the point of maximum curvature.
- The magnetization curve was found to be of the same general form as that of a paramagnetic metal, and gave indications that with a sufficient force magnetic saturation would probably be attained.
- Rhoads obtained a cyclic curve for iron which indicated thermo-electric hysteresis of the kind exhibited by Nagaoka's curves for magnetic strain.
- He also experimented with nickel and again found a resemblance to the strain curve.
- 1° S., where it runs close to the upper waters of the Athi, flows in a wide curve N.E., nearly reaching the equator.
- 3 the curve of brightness for one component is ABC, and for the other OA'C'; and the curve representing half the combined brightnesses is E'BE.
- Such a curve is shown in fig.
- The origin of co-ordinates 0 corresponds to v = 0; and the asymptotic points J, J', round which the curve revolves in an ever-closing spiral, correspond to v= =co .
- For the osculating circle at any point includes the whole of the y curve which lies beyond; and the successive convolutions envelop one another without intersection.
- The utility of the curve depends upon the fact that the elements of arc represent, in amplitude and phase, the component vibrations due to the corresponding portions of the primary wave-front.
- If the slit is of 'constant width and we require the illumination at various points on the screen behind it, we must regard the arc of the curve as of constant length.
- Yet another fashion was that adopted by the flamens, who passed the right-hand portion of the toga over the right shoulder and arm and back over the left shoulder, so that it hung down in a curve over the front of the body; the upper edge was folded over.
- If a point be in motion in any orbit and with any velocity, and if, at each instant, a line be drawn from a fixed point parallel and equal to the velocity of the moving point at that instant, the extremities of these lines will lie on a curve called the hodograph.
- At first it was known by the Dutch simply as the "fuyck" (hoop), from the curve in the river at this point, whence was soon derived the name Beverfuyck or Beverwyck.
- They then form part of a system of ranges which curve north of the sources of the Chindwin river, and with the Kumon range and the hills of the Jade and Amber mines, make up a highland tract separated from the great Northern Shan plateau by the gorges of the Irrawaddy river.
- West of this line the rocks are chiefly Tertiary and Quaternary; east of it they are mostly Palaeozoic or gneissic. In the western mountain ranges the beds are thrown into a series of folds which form a gentle curve running from south to north with its convexity facing westward.
- The resultant horizontal thrust in any direction is obtained by drawing parallel horizontal lines round the boundary, and intersecting a plane perpendicular to their direction in a plane curve; and then investigating the thrust on this plane area, which will be the same as on the curved surface.
- Then 4, =o over the cylinder r = a, which may be considered a fixed post; and a stream line past it along which 4, = Uc, a constant, is the curve (r - ¢2) sin 0=c, (x2 + y2) (y - c) - a 2 y = o, (3) a cubic curve (C3).
- A relative stream line, along which 1/,' = Uc, is the quartic curve y-c=?![2a(r-x)], x = 4a2y2-(y g)4, r- 4a2y2 +(y c) 4, 7) 4 a (y-c) 4a(y and in the absolute space curve given by 1', dy= (y- c)2, x= 2ac_ 2a log (y -c) (8) 2ay y - c 34.
- Motion symmetrical about an Axis.-When the motion of a liquid is the same for any plane passing through Ox, and lies in the plane, a function ' can be found analogous to that employed in plane motion, such that the flux across the surface generated by the revolution of any curve AP from A to P is the same, and represented by 2s-4 -11'o); and, as before, if d is the increase in due to a displacement of P to P', then k the component of velocity normal to the surface swept out by PP' is such that 274=2.7ryk.PP'; and taking PP' parallel to Oy and Ox, u= -d/ydy, v=dl,t'/ydx, (I) and 1P is called after the inventor, " Stokes's stream or current function," as it is constant along a stream line (Trans.
- If through every point of a small closed curve the vortex lines are drawn, a tube is obtained, and the fluid contained is called a vortex filament.
- Hence in any infinitesimal part of the fluid the circulation is zero round every small plane curve passing through the vortex line; and consequently the circulation round any curve drawn on the surface of a vortex filament is zero.
- = -dQ+1dg2, and integrating round a closed curve (udx+vdy+wdz) =0, and the circulation in any circuit composed of the same fluid particles is constant; and if the motion is differential irrotational and due to a velocity function, the circulation is zero round all reconcilable paths.
- The circulation being always zero round a small plane curve passing through the axis of spin in vortical motion, it follows conversely that a vortex filament is composed always of the same fluid particles; and since the circulation round a cross-section of a vortex filament is constant, not changing with the time, it follows from the previous kinematical theorem that aw is constant for all time, and the same for every cross-section of the vortex filament.
- The quadratrix of Dinostratus was well known to the ancient Greek geometers, and is mentioned by Proclus, who ascribes the invention of the curve to a contemporary of Socrates, probably Hippias of Elis.
- Dinostratus, a Greek geometer and disciple of Plato, discussed the curve, and showed how it effected a mechanical solution of squaring the circle.
- From every point of the curve of intersection, perpendiculars are drawn to the axis.
- The intercept on the axis of y is 2a/7r; therefore, if it were possible to accurately construct the curve, the quadrature of the circle would be effected.
- The curve also permits the solution of the problems of duplicating a cube and trisecting an angle.
- The vapour pressure-composition curve will now be concave to the axis of composition, the minima corresponding to the pure components.
- The vapour tension may approximate to a linear function of the composition, and the curve will then be practically a straight line.
- If we determine the freezing-points of a number of mixtures varying in composition from pure A to pure B, we can plot the freezing-point curve.
- In such a curve the percentage composition can be plotted horizontally and the temperature of the freezing-point vertically, as in fig.
- In the case of two substances which neither form compounds nor dissolve each other in the solid state, the complete freezing-point curve takes the form shown infig.5.
- The freezingpoint curve sometimes indicates the existence of chemical compounds.
- 6, consisting of two downward sloping branches meeting in the eutectic point, and that of thallium and tin, the upper curve of fig.
- 7 gives the freezing-point curve of mercury and thallium; here A and E are the melting-points of pure mercury and pure thallium, and the branches AB and ED do not cut each other, but cut an intermediate rounded branch BCD.
- Sometimes a freezing-point curve contains more than one intermediate summit, so that more than one compound is indicated.
- For example, in the curve for gold-aluminium, ignoring minor singularities, we find two intermediate summits, one at the percentage Au 2 A1, and another at the percentage AuAl 2.
- In the curve for sodium-cadmium, the compound NaCd 2 is plainly shown.
- We must not take it for granted, when the freezing-point curve gives no indication of the compound, that the compound does not exist in the solid alloy.
- For example, the compound Cu3Sn is not indicated in the freezing-point curve, and indeed a liquid alloy of this percentage does not begin to solidify by the formation of crystals of Cu 3 Sn; the liquid solidifies completely to a uniform solid solution, and only at a lower temperature does this change into crystals of the compound, the transformation being accompanied by a considerable evolution of heat.
- It is evident that any other property can be represented by similar diagrams. For example, we can construct the curve of conductivity of alloys of two metals or the surface of conductivity of ternary alloys, and so on for any measurable property.
- This may be interpreted as the equation of the border curve giving the relation between p and 0, but is more easily obtained by considering the equilibrium at constant pressure instead of constant volume.
- To find the border curve of equilibrium between the two states, giving the saturation pressure as a function of the temperature, we have merely to equate the values of G and G".
- Two of these figures stood at the end of a re-entrant curve, several pieces of which are preserved.
- As an additional claim to remembrance, he was the first to solve Leibnitz's problem of the isochronous curve (Acta Eruditorum, 1690).
- He proposed the problem of the catenary or curve formed by a chain suspended by its two extremities, accepted Leibnitz's construction of the curve and solved more complicated problems relating to it.
- He determined the "elastic curve," which is formed by an elastic plate or rod fixed at one end and bent by a weight applied to the other, and which he showed to be the same as the curvature of an impervious sail filled with a liquid (lintearia).
- Among these were the exponential calculus, and the curve called by him the linea brachistochrona, or line of swiftest descent, which he was the first to determine, pointing out at the same time the relation which this curve bears to the path described by a ray of light passing through strata of variable density.
- Newton defined the diameter of a curve of any order as the locus of the centres of the mean distances of the points of intersection of a system of parallel chords with the curve; this locus may be shown to be a straight line.
- The normal vertical distribution of temperature is illustrated in curve A of fig.
- Curve B shows the typical distribution of temperature in an enclosed sea, in this case the Sulu Basin of the Malay Sea, where from the level of the barrier to the bottom the temperature remains uniform or homothermic. Curve C shows a typical summer condition in the polar seas, where layers of sea-water at different temperatures are superimposed, the arrangement from the surface to 200 fathoms is termed FIG.
- The tubs are usually formed into sets of from 2 to 12, the front one being coupled up by a short length of chain to a clamping hook formed of two jaws moulded to the curve of the rope which are attached by the " run rider," as the driver accompanying the train is called.
- Apparatus is added to some dynamometers by means of which a curve showing the variations of P on a distance base is drawn automatically, the area of the diagram representing the work done; with others, integrating apparatus is combined, from which the work done during a given interval may be read off directly.
- 12 times the area of the generating circle; the length of the curve is 8a.
- As for the marvels of Peru, the walls of the temple of the sun in Cuzco, with their circular form and curve inward, from the ground upward, are most imposing.
- The tangent scale moved freely in a socket fixed to the gun; its lower end rested on one of the cams, cut to a correct curve.
- Its white stone houses form a long curve between the uplands of Salisbury Plain,which sweep away towards the north and east, and the tract of park and meadow land lying south and west.
- In Hippotragus the stout and thickly ringed horns rise vertically from a ridge above the eyes at an obtuse angle to the plane of the lower part of the face, and then sweep backwards in a bold curve; while there are tufts of long white hairs near the eyes.
- 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.
- (a) The nature of the bounding curve or surface may not be exactly known, so that certain assumptions have to be made, a formula being then used which is adapted to these assumptions.
- The application of Simpson's rule, for instance, to a plane figure implies certain assumptions as to the nature of the bounding curve.
- 0 =L.z.6, where M' is the moment of the original curve with regard to the axis, L is the total length of the original curve, and š is the distance of the centroid of the curve from the axis.
- Similarly, analytical plane geometry deals with the curve described by a point moving in a particular way, while analytical plane mensuration deals with the figure generated by an ordinate moving so that its length varies in a particular manner depending on its position.
- The most simple case is that in which the trapezette tapers out in such a way that the curve forming its top has very close contact, at its extremities, with the base; in other words, the differential coefficients u', u", u"',.
- According to Proclus, a man named Hippias, probably Hippias of Elis (c. 460 B.C.), trisected an angle with a mechanical curve, named the quadratrix.
- We can represent waves of longitudinal displacement by a curve, and this enables us to draw very important conclusions in a very simple way.
- If this is done for every point we obtain a continuous curve Apbqcrd, which represents the displacement at every point at the given instant, though by a length at right angles to the actual displacement and on an arbitrary scale.
- In ordinary sound waves the displacement is very minute, perhaps of the order 105 cm., so that we multiply it perhaps by ioo,000 in forming the displacement curve.
- If the waves are continuous and each of the same shape they form a " train," and the displacement curve repeats itself.
- At J the displacement is forward, but since the curve at Q is parallel to the axis the displacement is approximately the same for all the points close to J, and the air is neither extended nor compressed, but merely displaced bodily a distance represented by JQ.
- The figure shows that when the curve of displacement slopes down in the direction of propagation there is compression, and the pressure is above the normal, and that when it slopes up there is extension, and the pressure is below the normal.
- If a wave travels on without alteration the travelling may be represented by pushing on the displacement curve.
- The distribution of velocity then is represented by the dotted curve and is forward when the curve is above the axis and Dackward when it is below.
- Generally, if any condition in the wave is carried forward unchanged with velocity U, the change of 4 at a given point in time dt is equal to the change of as we go back along the curve a distance dx = Udt at the beginning of dt.
- When the value of dyldx is not very small E is no longer constant, but is rather greater in compression and rather less in extension than -yP. This can be seen by considering that the relation between p and is given by a curve and not by a straight line.
- If both vibrate, the point describes a curve which appears continuous through the persistence of the retinal impression.
- The simplest form of wave, so far as our sensation goes - that is, the one giving rise to a pure tone - is, we have every reason to suppose, one in which the displacement is represented by a harmonic curve or a curve of sines, y=a sin m(x - e).
- The form of the curve is evidently as represented in fig.
- In this curve ABCD are nodes.
- The maximum height of the curve HM =a is the amplitude.
- If we transfer 0 to A, e=o, and the curve may be represented by y=a sin A x.
- If now the curve moves along unchanged in form in the direction ABC with uniform velocity U, the epoch e =OA at any time t will be Ut, so that the value of y may be represented as 2 y=a sin T (x - Ut).
- The chief experimental basis for supposing that a train of longitudinal waves with displacement curve of this kind arouses the sensation of a pure tone is that the more nearly a source is made to vibrate with a single simple harmonic motion, and therefore, presumably, the more nearly it sends out such a harmonic train, the more nearly does the note heard approximate to a single pure tone.
- Let it be represented by a displacement curve Ahbkc. Its periodicity implies that after a certain distance the displacement curve exactly repeats itself.
- Let ABCD be drawn at such level that the areas above and below it are equal; then ABCD is the axis of the curve.
- Since the curve represents a longitudinal disturbance in air it is always continuous, at a finite distance from the axis, and with only one ordinate for each abscissa.
- Fourier's theorem asserts that such a curve may be built up by the superposition, or addition of ordinates, of a series of sine curves of wave-lengths X, IX, 3A, 4A...
- Now we can see that two notes of the same pitch, but of different quality, or different form of displacement curve, will, when thus analysed, break up into a series having the same harmonic wave-lengths; but they may differ as regards the members of the series present and their amplitudes and epochs.
- Now we may resolve these trains by Fourier's theorem into harmonics of wave-lengths X, 2X, 3A, &c., where X=2AB and the conditions as to the values of y can be shown to require that the harmonics shall all have nodes, coinciding with the nodes of the fundamental curve.
- (25) The sum of the disturbance is obtained by adding (24) and (25) y = y l +y 2 = 2a cos Ut s i n 57 x, (26) At any given instant t this is a sine curve of amplitude 2a cos (27r/A)Ut, and of wave-length A, and with nodes at x = o, a A, A, ..., that is, there is no displacement at these nodes whatever the value of t, and between them the displacement is always a sine curve, but of amplitude varying between +2a and - 2a.
- The ordinate of the curve changes sign as we pass through a node, so that successive sections are moving always in opposite directions and have opposite displacements.
- The dotted curve represents the superposition, which simply doubles each ordinate.
- 26 (2), the dotted curve representing (I) tfr FIG.
- The tangent to the displacement curve is always parallel to the axis, that is, for a small distance the successive particles are always equally displaced, and therefore always occupy the same volume.
- If the plane does not contain the centre, the curve of intersection is a "small circle," and the solid cut off is a "segment."
- The curve of the main arch is a parabola.
- The lower portion of the figure is the curve of bending moments under the leading load.
- Till W 1 has advanced a distance a only one load is on the girder, and the curve A"F gives bending moments due to W 1 only; as W1 advances to a distance a+b, two loads are on the girder, and the curve FG gives moments due to W 1 and W2.
- The heavy continuous line gives the last-mentioned curve for the reverse direction of passage of the loads.
- With short bridges it is best to draw the curve of maximum bending moments for some assumed typical set of loads in the way just described, and to design the girder accordingly.
- But very great accuracy in drawing this curve is unnecessary, because the rolling stock of railways varies so much that the precise magnitude and distribution of the loads which will pass over a bridge cannot be known.
- Now, except for very short bridges and very unequal loads, a parabola can be found which includes the curve of maximum moments.
- This parabola is the curve of maximum moments for a travelling load uniform per ft.
- Hence the moment of the load on Am at C is wy0m, and the moment of a uniform load over any portion of the girder is w X the area of the influence curve Ip' G' E ' under that portion.
- This curve is a parabola.
- Consider any other point F of the curve, fig.
- 70, at a distance x from the vertex, the horizontal component of the resultant (tangent to the curve) will be unaltered; the vertical component V will be simply the sum of the loads between 0 and F, or wx.
- 72 with arcs of the length 1,, l2, l3, &c., and with the radii r1, r 2, &c. (note, for a length 2l 1 at each end the radius will be infinite, and the curve must end with a straight line tangent to the last arc), then let v be the measured deflection of this curve from the straight line, and V the actual deflection of the bridge; we have V = av/b, approximately.
- This method distorts the curve, so that vertical ordinates of the curve are drawn to a scale b times greater than that of the horizontal ordinates.
- The vertical distortion of the curve must not be so great that there is a very sensible difference between the length of the arc and its chord.
- The old Austria was very richly provided with raw materials; the coal and iron supply was especially rich; in the years immediately preceding the war the production of these two commodities followed in general a rising curve.
- These ideas are further developed in various papers in the Bulletin and in his L'Anthropometrie, ou mesure des differentes facultes de l'homme (18'ji), in which he lays great stress on the universal applicability of the binomial law, - according to which the number of cases in which, for instance, a certain height occurs among a large number of individuals is represented by an ordinate of a curve (the binomial) symmetrically situated with regard to the ordinate representing the mean result (average height).
- Thus the path of the ray when the aether is at rest is the curve which makes fds/V least; but when it is in motion it is the curve which makes fds/(V+lug-m y -I-nw) least, where (l,m,n) is the direction vector of Ss.
- 8 and 9, in a curve showing the relation between p and D the gravimetric density, which is the specific gravity of the P lb of powder when filling the volume C, cub.
- In the same work Wallis obtained an expression for the length of the element of a curve, which reduced the problem of rectification to that of quadrature.
- At Montrejeau it receives on the left the Neste, and encountering at this point the vast plateau of Lannemezan is forced to turn abruptly east, flowing in a wide curve to Toulouse.
- It follows exactly the curve of the mainland, and is continued into Panama, under the name of the Cordillera de Chiriqui.
- The interior curve formed by the Gulf of Mexico is comparatively regular and has a coast-line of about 1400 m.
- The outer curve facing the Pacific is less regular, is deeply broken by the Gulf of California, and has a coast-line of 4574 m., including that of the Gulf.
- The principal watershed is formed by the sierras of the state of Mexico, from which streams flow north-east to the Gulf of Mexico, northwest to the Pacific and south-west to the same coast below its great eastward curve.
- The apical end of the rotifer usually narrows suddenly beyond the curve of the gut and the cloacal aperture to form the foot of pseudopodium which ends in an organ of attachment, a pair of movable toes, each with the opening of a cement-gland (gl) at its tip. Thus for orientation we place the rotifer like the cuttle-fish, head downwards: the ciliated disk is basal or oral, proximal to the rest of the animal, the foot is apical, and the brain and cloacal aperture are anterodorsal.
- The form of the limacon depends on the ratio of the two constants; if a be greater than b, the curve lies entirely outside the circle; if a equals b, it is known as a cardioid; if a is less than b, the curve has a node within the circle; the particular case when b= 2a is known as the trisectrix.
- Some of these reverted breeds have developed horns of considerable size, although not showing that regularity of curve distinctive of the wild race.
- The horns of the males are very large, and curve round after the manner of the wild goat, with a tuft of hair between and in front.
- The swiftest current te,-ids, by reason of centrifugal force, to follow the outer side of every significant curve in the channel; hence the concave bank, against which the rapid current sweeps, is worn away; thus any chance irregularity is exaggerated, and in time a series of large serpentines or meanders is developed,, the most-symmetrical examples at present being those near Greenville, Miss.
- This low range turns westward in a curve through the Rattlesnake Mountains towards the high Wind River Mountains (Gannett Peak, 3,775 ft.), an anticlinal range within the body of the mountain system, with flanking strata rising well on the slopes.
- The upper canines especially are of great size, and curve outwards, forwards and upwards.
- The curve itself is sometimes termed the " circumference."
- Any line through the centre and terminated at both extremities by the curve, e.g.
- The general equation to the circle in trilinear co-ordinates is readily deduced from the fact that the circle is the only curve which intersects the line infinity in the circular points.
- Apollonius' genius takes its highest flight in Book v., where he treats of normals as minimum and maximum straight lines drawn from given points to the curve (independently of tangent properties), discusses how many normals can be drawn from particular points, finds their feet by construction, and gives propositions determining the centre of curvature at any point and leading at once to the Cartesian equation of the evolute of any conic.
- Entering the department of Gers, the Adour receives the Arros on the right bank and begins to describe the large westward curve which takes it through the department of Landes to the sea.
- The waterside streets, however, follow the curve of the beach.
- Both upper and lower incisors are regularly curved, the upper ones: slightly more so than the lower; and, their growth being continuous, should anything prevent the normal wear by which their length is regulated - as by the loss of one of them, or by displacement owing to a broken jaw or other cause - the unopposed incisor may gradually curve upon itself until a complete circle or more has, been formed, the tooth sometimes passing through some part of the animal's head.
- A spectroscope may be compared to a mechanical harmonic analyser which when fed with an irregular function of one variable represented by a curve supplies us with the sine curves into which the original function may be resolved.
- Proposition 30 describes the construction of a curve of double curvature called by Pappus the helix on a sphere; it is described by a point moving uniformly along the arc of a great circle, which itself turns about its diameter uniformly, the point describing a quadrant and the great circle a complete revolution in the same time.
- The area of the surface included between this curve and its base is found - the first known instance of a quadrature of a curved surface.
- In the same preface is included (a) the famous problem known by Pappus's name, often enunciated thus: Having given a number of straight lines, to find the geometric locus of a point such that the lengths of the perpendiculars upon, or (more generally) the lines drawn from it obliquely at given inclinations to, the given lines satisfy the condition that the product of certain of them may bear a constant ratio to the product of the remaining ones; (Pappus does not express it in this form but by means of composition of ratios, saying that if the ratio is given which is compounded of the ratios of pairs - one of one set and one of another - of the lines so drawn, and of the ratio of the odd one, if any, to a given straight line, the point will lie on a curve given in position), (b) the theorems which were rediscovered by and named after Paul Guldin, but appear to have been discovered by Pappus himself.
- The great chain of volcanoes which runs through Sumatra and Java is continued eastwards into the Moluccas, and terminates in a hooklike curve which passes through the Damar Islands to the Banda group. Outside this hook lies a concentric arc of non-volcanic islands, including Tenimber, the Lesser Kei Islands, Ceram and Buru; and beyond is still a third concentric arc extending from Taliabu to the Greater Kei Islands.
- On the whole it appears that the older rocks are found more particularly towards the interior of the curve, and the newer rocks towards the exterior.
- The angle between a line and a curve (mixed angle) or between two curves (curvilinear angle) is measured by the angle between the line and the tangent at the point of intersection, or between the tangents to both curves at their common point.
- Newton called attention to the fact that a falling body moves in a curve, diverging slightly from the plumb-line vertical.
- If all the ice be melted, we pass along the vapour pressure curve of water OA.
- If all the water be frozen, we have the vapour pressure curve of ice OB; while, if the pressure be raised, so that all the vapour vanishes, we get the curve OC of equilibrium between the pressure and the freezing point of water.
- If the supply of ice fails first the temperature will rise, and, since solid salt remains, we pass along a curve OA giving the relation between temperature and the vapour pressure of the saturated solution.
- If, on the other hand, the salt of the cryohydrate fails before the ice the water given by the continued fusion dilutes the solution, and we pass along the curve OB which shows the freezing points of a series of solutions of constantly increasing dilution.
- Or, by increasing the pressure, we eliminate the vapour and obtain the curve OF giving the relation between pressure, freezing point and composition when a saturated solution is in contact with ice and salt.
- Experiments on the relation between temperature and concentration are illustrated by projecting the curve OA of fig.
- The pressure at each point should be that of the vapour, but since the solubility of a solid does not change much with pressure, measurements under the constant atmospheric pressure give a curve practically identical with the theoretical one.
- The curve OP represents the varying solubility of the hydrate as the temperature rises from the cryohydric point to 32.6°.
- When this process is complete the temperature rises, and we pass along a new curve giving the equilibrium between anhydrous crystals, solution and vapour.
- 4 A represents the freezing point of pure water, and AB the freezing point curve showing the depression of the freezing point as phenol is added.
- 5 freezing point curve, and then travels along it till the non-variant point is reached.
- In this way two temperature points are obtained in the investigation - the higher giving a point on the equilibrium curve, the lower showing the non-variant point.
- Proceeding along the curve in either direction, we come to a non-variant or eutectic point.
- Thus in interpreting complicated freezing point curves, we must look for chemical compounds where the curve shows a maximum, and for a eutectic or cryohydrate where two curves meet at a minimum point.
- The experi mental curve of solubility is shown in fig.
- At A we 66 have the freezing point of pure water, which is lowered by the gradual addition of 46 ferric chloride in the manner shown by the curve AB.
- In a similar way the curve FGH, between 30° and 55°, shows the effect of the hydrate Fe2Cl6.5H20, and the curve HJK that of the hydrate Fe 2 C1 6.4H 2 O, which, when pure, melts at 73.5° - the point J on the diagram.
- At the point K, 66°, begins the solubility curve of the anhydrous salt, Fe2C16, the fusion point of which when pure is beyond the limits of the diagram.
- When the curve BC is reached, Fe 2 C1 6 - 12H 2 0 separates out, and the solution solidifies.
- Further renewal of water will cause first liquefaction, as the curve CD is passed, and then resolidification to Fe 2 C1 6.7H 2 0 when DE is cut.
- But at intermediate compositions we can only guess at the form of the energy-composition curve, and the freezing point composition curve, deduced from it, will vary according to the supposition which we make.
- It will be noticed that in all these theoretical curves the points of initial fusion and solidification do not in general coincide; we reach a different curve first according as we approach the diagram from below, where all is solid, or from above, where all is liquid.
- The second and third figures, too, show that the presence of solid solutions may simulate the phenomena of chemical combination, where the curve reaches a maximum, and of non-variant systems where we get a minimum.
- All available evidence, from the freezing point curve and from other sources must be scrutinized before an opinion is pronounced.
- If the temperature at which this dense spontaneous shower of crystals is found be determined for different concentrations of solution, we can plot a "supersolubility curve," which is found generally to run roughly parallel to the "solubility curve" of steady equilibrium between liquid and already existing solid.
- 6 When the temperature sank to a, on the freezing point curve, E40 crystals of pure A would appear.
- The residual liquid would thus become richer in B, and the tem perature and composition would pass along the curve till E, the eutectic point, was reached.
- But, if no solid be present initially, or if the cooling be rapid, the liquid of composition x becomes supersaturated and may cool till the supersaturation curve is reached at b, and a cloud of A crystals comes down.
- The conditions may then remain those of equilibrium along the curve f E, but before reaching f the solution may become supersaturated with B and deposit B crystals spontaneously.
- 15 the curve I represents Boyle's law if the volume is taken to be that of the solution, and the curve II if the volume is that of the solvent.
- Even this correction is not sufficient in solution of sugar, where the theoretical curve II lies below the experimental observations.
- The apparently strange and inconvenient position of the Stadium relatively to the Altis was due simply to the necessity of obeying the conditions of the ground, here determined by the curve of the loweslopes which bound the valley on the north.
- A neat rainguard made of sheet metal, to the same curve as the body of the periscope and almost 8 inches long, is attached to the upper prism box by two spring straps.
- On the landward side, Braila has the shape of a crescent, the curve of its outer streets following the line of the old fortifications, dismantled in 1829.
- The curve is symmetrical about the line x = y, and consists of two infinite branches asymptotic to the line x+y+a = o and a loop in the first quadrant.
- It may be traced by giving m various values in the equations x=3am/ ('1-1-m 3 ),' y=3am2 (1-1-m 3), since by eliminating m between these relations the equation to the curve is obtained.
- The area of the loop, which equals the area between the curve and its asymptote, is 3a/2.
- Hence the main watershed extends eastwards, culminating in the Aiguille de Scolette (11,500 ft.), but makes a great curve to the north-west and back to the south-east before rising in the Rochemelon (11,605 ft.), which may be considered as a re-entering angle in the great rampart by which Italy is guarded from its neighbours.
- If the two hands be placed flat upon the table, in the angle between the two books, and the cloth pushed towards the corner, it will at once be rucked up into a fold which will follow a curve not unlike that of the Alps.
- The half-fan is a combination of the two forms, but as regards pruning does not materially differ from the horizontal, as two opposite side branches are produced in succession upwards till the space is filled, only they are not taken out so abruptly, but are allowed to rise at an acute angle and then to curve into the horizontal line.
- Potatoes are cultivated in all the provinces, but especially in the Palatinate and in the Spessart district, which lies in the north-west within a curve of the Main.
- The coast-line extends in a double curve from south-west to northeast, and is formed by a row of sand dunes, 171 m.
- The Maas, whose course is almost parallel to that of the Rhine, follows in a wide curve the general slope of the country, receiving the Roer, the Mark and the Aa.
- In many Basidiomycetes minute branches arise below the septa; their tips curve over the outside of the latter, and fuse with the cell above just beyond it, forming a clamp-connexion.
- The city faces upon a curve in the river bank forming what is called the Bay of Asuncion, and is built on a low sandy plain, rising to pretty hillsides overlooking the bay and the low, wooded country of the Chaco on the opposite shore.'
- The Bug, another right-hand tributary of the Vistula, describes a wide curve concentric with those of the middle Vistula and the Narew, and separates the Polish governments of Lublin and Siedlce from the Russian governments of Volhynia and Grodno.
- Again, the July isotherm of 75° crosses the middle section of the Caspian, nearly coinciding with the January isotherm of 25°, while that of 80° skirts the southern shore of the sea, nearly coinciding with the January curve of 40°, so that the mean annual range over the northern section of the sea is 60° and over the southern section 40°.
- 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.
- In this case the curve representing the distribution of temperature is a parabola, and the conductivity k is deduced from the mean rise of temperature (R-R°)/aR° by observing the increase of resistance R-R° of the bar, and the current C. It is also necessary to measure the cross-section g, the length 1, and the temperature-coefficient a for the range of the experiment.
- The cycloid was a famous curve in those days; it had been discussed by Galileo, Descartes, Fermat, Roberval and Torricelli, who had in turn exhausted their skill upon it.
- Pascal solved the hitherto refractory problem of the general quadrature of the cycloid, and proposed and solved a variety of others relating to the centre of gravity of the curve and its segments, and to the volume and centre of gravity of solids of revolution generated in various ways by means of it.
- Those of the upper jaw are directed upwards from their bases, so that they never enter the mouth, but pierce the skin of the face, thus resembling horns rather than teeth; they curve backwards, downwards, and finally often forwards again, almost or quite touching the forehead.
- Points on the same curve are supposed to have the same average number of auroras in the year, and this average number is shown adjacent to the curve.
- Between the Shetlands and Iceland we cross the curve of maximum frequency, and farther north the frequency diminishes.
- Greenland lies to the north of Fritz's curve of maximum auroral frequency, and the suggestion has been made that the zone of maximum frequency expands to the south as sun-spots increase, and contracts again as they diminish, the number of auroras at a given station increasing or diminishing as the zone of maximum frequency approaches to or recedes from it.
- The curve of the arch turns in slightly below the springing, giving a horse-shoe shape.
- There are many local irregularities, but the general direction is maintained as far as the southern extremity of Greece, where the folds show a tendency to curve towards Crete.
- She invented and discussed the curve known as the "witch of Agnesi" (q.v.) or versiera.
- The ordinate of the dotted curve which contains its "centre of gravity" has, of course, for its abscissa the "mean" number of glands; the maximum ordinate of the curve is, however, at 2.98, or sensibly at 3 glands, showing what Pearson has called the "modal" number of glands, or the number occurring most frequently.
- The ordinate which divides the area of the dotted curve into two equal areas is the median of Galton: it lies in this case nearly at 3.38 glands.
- Trans., A., 1893) that this frequency may be closely represented by the curve whose equation is y = O.21 122 5 x-( 332 (7.3 2 53 - x) 3.142.
- The curve, and the observations it represents, are drawn in fig.
- 3° 14' S., and has an outward curve.
- The form of the curve is shown in fig.
- 1, where P is a point on the curve equidistant from the fixed line AB, known as the directrix, and the fixed point F known as the focus.
- Then EG produced meets FP in a point on the curve.
- The cartesian parabola is a cubic curve which is also known as the trident of Newton on account of its three-pronged form.
- John Wallis utilized the intersections of this curve with a right line to solve cubic equations, and Edmund Halley solved sextic equations with the aid of a circle.
- As the two lesser roots are made more and more equal the oval shrinks in size and ultimately becomes a real conjugate point, and the curve, the equation of which is y2= (x - a) 2 (x - b) (in which a > b) consists of this point and a bell-like branch resembling the right-hand member of fig.
- 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.
- (See CURVE; and GEOMETRY, ANALYTICAL.)
- Some of these are united to the mainland and to each other by jetties which curve round so as to form the Port de Refuge, a haven available only in fair weather.
- Large flowing leaves of acanthus and other plants were beaten out with wonderful spirit and beauty of curve.
- The length of India from north to south, and its greatest breadth from east to west, are both about 1900 m.; but the triangle tapers with a pear-shaped curve to a point at Cape Comorin, its southern extremity.
- At this north-western angle (in 35° N., 74° E.) the mountains curve southwards, and India is separated by the well-marked ranges of the Safed Koh and Suliman from Afghanistan; and by a southern continuation of lower hills from Baluchistan.
- 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.
- Stracciati By The Method Of Mixture Between O° And 30° C., Though Their Curve Is Otherwise Similar To Rowland'S, Had Appeared To Indicate A Minimum At 20° C., Followed By A Rapid Rise.
- The Specific Heat Itself Can Be Deduced Only By Differentiating The Curve Of Observation, Which Greatly Increases The Uncertainty.
- In both cases the rise to a maximum is more rapid than the decline to a minimum, and in fact some of the minor peculiarities of the sunspot curve are closely imitated by the light-curves of variable stars.
- Either of the two is the all, as, for example, the law of the convexity of the curve is the law of the curve and the law of its concavity.
- But by a most skilful manoeuvre Narses contrived to draw his lines into a curve, so that his mounted archers on each flank could aim their arrows at the backs of the troops who formed the other side of the Alamannic wedge.
- The width of the photographic sheet which receives the spot of light reflected from the mirrors in the above instruments is generally so great that in the case of ordinary changes the curve does-not go off the paper.
- Cady, Terrestrial Magnetism, 1904, 9, 69, describing a declination magnetograph in which the record is obtained by means of a pen acting on a moving strip of paper, so that the curve can be consulted at all times to see whether a disturbance is in progress.
- The curve shows two rises, one at the beginning of winter, and the other at the commencement of the monsoon, and at both these times the people are driven indoors.
- In the east a well-defined mountain system runs nearly parallel to the Black Sea coast from Batum to Sinope, forming a gentle curve with its convexity facing southwards.
- The same holds for the four points B, C, D, E and so on; but since a parabola is uniquely determined by the direction of its axis and by three points on the curve, the successive parabolas ABCD, BCDE, CDEF ...
- For example, in the case of a particle lying on a smooth curve, or on a smooth surface, if it be displaced along the curve, or on the surface, the virtual work of the normal component of the pressure may be ignored, since it is of the second order.
- It is assumed that the form can be sufficiently represented by a plane curve, that the stress (tension) at any point P of the curve, between the two portions which meet there, is in the direction of the tangent at P, and that the forces on any linear element s must satisfy the conditions of equilibrium laid down in I.
- We will suppose in the first instance that the curve is plane.
- -- T.r-OT If T, T + aT be the tensions at P, Q, and 4 be the angle between the directions of the curve at these points, the components Q of the tensions along the tangent at P give (T + T) cos T,
- Suppose, for example, that we have a light string stretched over a smooth curve; and let Rs denote the normal pressure (outwards from the centre of curvature) on bs.
- Next suppose that the curve is rough; and let Fas be the tangential force of friction on s.
- Again, take the case of a string under gravity, in contact with a smooth curve in a vertical plane.
- This is the intrinsic equation of the curve.
- 0 is at a distance a vertically below the lowest point (4 = 0) of the curve.
- It follows that the osculating plane of the curve formed by the string must contain the normal to the surface, i.
- The relation between x and t in any particular case may be illustrated by means of a curve constructed with I as abscissa and x as ordinate.
- This is called the curve of positions or space-time curve; its gradient represents the velocity.
- A curve with I as abscissa and u as ordinate is called the curve of velocities or velocity-time curve.
- The curve of positions corresponding to (4) is a parabola, and that of velocities is a straight line.
- 62 shows the curves of position and velocity; they both have the form of the curve of sines.
- It may be noticed that if the scales of x and be properly adjusted, the curve of positions in the present problem is the portion of a cycloid extending from a vertex to a cusp.
- Take, for example, the case of a particle moving on a smooth curve in a vertical plane, under the action of gravity and the pressure R of the curve.
- In the case of the pendulum the tension of the string takes the place of the pressure of the curve.
- Moreover, the case n=2 is the only one in which the critical orbit (27) can be regarded as the limiting form of a closed curve.
- The herpolhode curve in the fixed plane is obviously confined between two concentric circles which it alternately touches; it is not in general a re-entrant curve.
- It has been shown by Dc Sparre that, owing to the limitation imposed on the possible forms of the momental ellipsoid by the relation B+C>A, the curve has no points of inflexion.
- The centres of pressure at the joints are also called centres of resistance, and the curve passing through these points is called a line of resistance.
- A curve tangential to all the sides of the polygon is the line of pressures.
- (8) A parallel projection of a curve, or of a surface of a given algebraical order, is a curve or a surface of the same order.
- The path of P is a curve of the kind called epitrochoids.
- To work with a wheel of any other figure, its section must be a rolling curve, subject to the condition that the perpendicular distance from the pole or centre of the wheel to a straight line parallel to the direction of the motion of the rack shall be constant.
- If the same rolling curve R, with the same tracing-point T, be rolled on the outside of any other pitch-circle, it will have the fare of a tooth suitable to work with the flank AT.
- In like manner, if either the same or any other rolling curve be rolled the opposite way, on the outside of the pitch-circle BB, so that the tracing point T shall start from A, it will trace the face AT of a tooth suitable to work with a flank traced by rolling the same curve R with the same tracing-point T inside any other pitch-circle.
- The figure of the path of con tact is that traced on a fixed plane by the tracing-point, when the rolling curve is rotated in such a manner as always to touch a fixed straight line EIE (or EIE, as the case may be) at a fixed point I (or I).
- This curve, bein~ an asymptote to its axis, is capable of being indefinitely proloi~ged towards X; but in designing pivots it should stop before the angle PTY becomes less than the angle of repose of the rubbing surfaces, otherwise the pivot will be liable to stick in its bearing.
- This equation gives a relation between the inclination of the curve to the horizon and the height above the level of the liquid.
- Resolving vertically we find that the weight of the liquid raised above the level must be equal to T(sin 0 2 - sin 01), and this is therefore equal to the area P 1 P 2 A 2 A 1 multiplied by gp. The form of the capillary surface is identical with that of the " elastic curve," or the curve formed by a uniform spring originally straight, when its ends are acted on by equal and T 2 opposite forces applied either to the ends themselves or to solid pieces attached to them.
- Drawings of the different forms of the curve may be found in Thomson and Tait's Natural Philosophy, vol.
- Let us first determine the nature of a curve, such that if it is rolled on the axis its origin will trace out the meridian section of the bubble.
- Since at any instant the rolling curve is rotating about the point of contact with the axis, the line drawn from this point of contact to the tracing point must be normal to the direction of motion of the tracing point.
- Hence if N is the point of contact, NP must be normal to the traced curve.
- Also, since the axis is a tangent to the rolling curve, the ordinate PR is the perpendicular from the tracing point P on the tangent.
- Hence the relation between the radius vector and the perpendicular on the tangent of the rolling curve must be identical with the relation between the normal PN and the ordinate PR of the traced curve.
- If A 1, C 1 are the disks, so that the distance between them is less than irr, the curve must be produced beyond the disks before it is at its mean distance from the axis.
- If, on the other hand, the disks are at A2 and C2, so that the distance between them is greater than 7rr, the curve will reach its mean distance from the axis before it reaches the disks.
- Beyond them a road branches to the right, sweeping round in a broad curve to the space in front of the temple of Apollo.
- The outer side of this curve is bounded by a row of treasuries, similar to those found at Delphi and Olympia, and serving to house the more costly offerings of various islands or cities.
- The curve obtained on joining the former points then brings out a number of facts, foremost among which are (1) that as long as the conditions remain constant the doubling periods - i.e.
- It was also shown that exposure to light, dilution or exhaustion of the food-media, the presence of traces of poisons or metabolic products check growth or even bring it to a standstill; and the death or injury of any single cell in the filamentous series shows its effect on the curve by lengthening the doubling period, because its potential progeny have been put out of play.
- - Curve of growth of a filament of Bacillus ramosus (Fraenkel), constructed from data such as in fig.
- 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.
- On the circumference of the earthern circle or surrounding rampart (not shown on plan), which is here intentionally broken, a great recumbent stone - the slaughter stone - lies along the axis: and across the axis, near the central curve of the inner horseshoe, lies a fine recumbent stone - the altar stone-15 ft.
- Since the circumference of a circle is proportional to its radius, it follows that if the ratio of the radii be commensurable, the curve will consist of a finite number of cusps, and ultimately return into itself.
- In the particular case when the radii are in the ratio of I to 3 the epicycloid (curve a) will consist of three cusps external to the circle and placed at equal distances along its circumference.
- Similarly, the corresponding epitrochoids will exhibit three loops or nodes (curve b), or assume the form shown in the curve; c. It is interesting to compare the forms of these curves with the three forms of the cycloid.
- The hypocycloid derived from the same circles is shown as curve d, and is seen to consist of three cusps arranged internally to the fixed circle; the corresponding hypotrochoid consists of a three-foil and is shown in curve e.
- This curve is the envelope of a line of constant length, which moves so that its extremities are always on two fixed lines at right angles to each other, i.e.of the line xla+y//= I, with the condition a 2 + 1 3 2 = I/a, a constant.
- The strata are thrown into folds which run in the direction of the mountain ridges, forming a curve with the convexity facing the south-east.
- If, again, the wing be suddenly elevated in a strictly vertical direction, as at c d, the wing as certainly darts upwards and forwards in a double curve to e, thus converting the vertical up strokes into an upward, oblique, forward stroke.
- Plinlimmon (2468 ft.) is the highest of the hills, and forms a sort of hydrographic centre for the group, as from its eastern base the Severn and the Wye take their rise - the former describing a wide curve to east and south, the latter forming a chord to the arc in its southward course.
- It appears as a series of rounded hills of no great elevation, running in a curve from the mouth of the Axe to Flamborough Head, roughly parallel with the Oolitic escarpment.
- The name cycloid is now restricted to the curve described when the tracing-point is on the circumference of the circle; if the point is either within or without the circle the curves are generally termed trochoids, but they are also known as the prolate and curtate cycloids respectively.
- Francis Schooten (Commentary on Descartes) assigns the invention of the curve to Rene Descartes and the first publication on this subject after Descartes to Marin Mersenne.
- Evangelista Torricelli, in the first regular dissertation on the cycloid (De dimensione cycloidis, an appendix to his De dimensione parabolae, 1644), states that his friend and tutor Galileo discovered the curve about 1599.
- John Wallis discussed both the history and properties of the curve in a tract De cycloide published at Oxford in 1659.
- Honore Fabri (Synopsis geometrica, 1669) treated of the curve and enumerated many theorems concerning it.
- Galileo attempted the evaluation by weighing the curve against the generating circle; this rough method gave only an approximate value, viz., a little less than thrice the generating circle.
- Blaise Pascal determined the area of the section made by any line parallel to the base and the volumes and centres of gravity of the solids generated by revolving the curve about its axis and base.
- The evaluation of the area of the curve had made Roberval famous in France, but Descartes considered that the value of his investigation had been grossly exaggerated; he declared the problem to be of an elementary nature and submitted a short and simple solution.
- The mechanical properties of the cycloid were investigated by Christiaan Huygens, who proved the curve to be tautochronous.
- When measured R from the vertex the results may be expressed in the forms s= 4a sin 20 and s = (8ay); the total length of the curve is 8a.
- 2, in o which the curve QOP is the evolute and QPR FIG.
- 1, curve c. The companion to the cycloid is a curve so named on account of its similarity of construction, form and equation to the common cycloid.
- The curve is shown in fig.
- The cartesian equation, referred to the fixed diameter and the tangent at B as axes may be expressed in the forms x= a6, y=a(I -cos 0) and y-a=a sin (x/afir); the latter form shows that the locus is the harmonic curve.
- This change of direction corresponds to a curve in the line of volcanic fissures which have contributed their products to the building of the islands.
- The simplest case of a caustic curve is when the reflecting surface is a circle, and the luminous rays emanate from a point on the circumference.
- Secondary caustics are orthotomic curves having the reflected or refracted rays as normals, and consequently the proper caustic curve, being the envelope of the normals, is their evolute.
- It is usually the case that the secondary caustic is easier to determine than the caustic, and hence, when determined, it affords a ready means for deducing, the primary caustic. It may be shown by geometrical considerations that the secondary caustic is a curve similar to the first positive pedal of the reflecting curve, of twice the linear dimensions, with respect to the luminous point.
- For a circle, when the rays emanate from any point, the secondary caustic is a limacon, and hence the primary caustic is the evolute of this curve.
- If the second medium be more highly refractive than the first, the secondary caustic is a hyperbola having the same focus and centre as before, and the caustic is the evolute of this curve.
- When the refracting curve is a circle and the rays emanate from any point, the locus of the secondary caustic is a Cartesian oval, and the evolute of this curve is the required diacaustic. These curves appear to have been first discussed by Gergonne.