Plane Sentence Examples

plane
  • I'll make a plane reservation.

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  • I'm still not going to watch your plane leave.

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  • The only thing left to do was pack and take a plane home.

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  • All right, I'll charter a plane for the morning.

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  • But he had flown to Arkansas on a plane with her.

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  • The little plane danced and swayed in the turbulence, constantly buffeted by the increas­ing wind.

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  • Alex was going to college when his parents died in the plane crash.

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  • I missed Betsy already and her plane hadn't left the ground.

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  • You'll help Julie to register Molly at the Keene school while I make the plane reservation.

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  • The plane was scheduled to leave in 45 minutes but one look at the departure board was indicative of things to come.

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  • To produce a medusa the actinula grows greatly along a plane at right angles to the vertical axis of the body, whereby the aboral surface of the actinula becomes the exumbrella, and the peristome becomes the subumbrella.

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  • He proved the law of the equilibrium on an inclined plane.

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  • If now we wish to represent the variations in some property, such as fusibility, we determine the freezing-points of a number of alloys distributed fairly uniformly over the area of the triangle, and, at each point corresponding to an alloy, we erect an ordinate at right angles to the plane of the paper and proportional in length to the freezing temperature of that alloy.

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  • Whatever this increased illumination may be, it can be precisely imitated by removing the mirror and placing a second lighted candle at the place occupied by the optical image of the first candle in the mirror, that is, as far behind the plane as the first candle was in front.

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  • If a long glass tube with plane ends, and containing some pellets of sodium is heated in the middle by a row of burners, the cool ends remain practically vacuous and do not become obscured.

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  • The normal position of this coil is with its plane parallel to the lines of force of the field.

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  • A specimen of one of these heavy glasses afterwards became historically important as the substance in which Faraday detected the rotation of the plane of polarization of light when the glass was placed in the magnetic field, and also as the substance which was first repelled by the poles of the magnet.

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  • The first evidence which he obtained of the rotation of the plane of polarization of light under the action of magnetism was on the 13th of September 1845, the transparent substance being his own heavy glass.

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  • The discovery of the magnetic rotation of the plane of polarized light, though it did not lead to such important practical applications as some of Faraday's earlier discoveries, has been of the highest value to science, as furnishing complete dynamical evidence that wherever magnetic force exists there is matter, small portions of which are rotating about axes parallel to the direction of that force.

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  • Galileo proceeded to measure the motion of a body on a smooth, fixed, inclined plane, and found that the law of constant acceleration along the line of slope of the plane still held, the acceleration decreasing in magnitude as the angle of inclination was reduced; and he inferred that a body, moving on a smooth horizontal plane, would move with uniform velocity in a straight line if the resistance of the air, and friction due to contact with the plane, could be eliminated.

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  • Immediately above this plane surface and almost touching it is a system of wires which enables angular distances from the centre of the field to be read at the eyepiece below.

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  • Stevinus was the first to show how to model regular and semiregular polyhedra by delineating their frames in a plane.

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  • In its simplest form it consists of a direct-vision spectroscope, having an adjustable slit (called "camera slit"), instead of an eyepiece, in the focal plane of the observing telescope.

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  • A cross-hair, in the focal plane of an eyepiece, is then moved horizontally until it coincides with the line in question.

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  • Another simple case is where the plane or slightly convex surface of the stroma rises at its margins and overgrows the sporogenous hyphal ends, so that the spores, asci, &c., come to lie in the depression of a cavity - e.g.

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  • This extraordinary diamond weighed 30254 carats (13 lb) and was clear and water white; the largest of its surfaces appeared to be a cleavage plane, so that it might be only a portion of a much larger stone.

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  • This was the first serious effort made in the United States to elevate secondary education to the plane on which it belonged.

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  • Its course should be as straight and as near a true inclined plane as possible.

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  • Here the con ductor should be led along the highest end or side of the meadow in an inclined plane; should it terminate in the meadow, its end should be made to taper when there are no feeders, or to terminate in a feeder.

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  • If all the masses lie in a plane (1=0) we have, in the notation of (25), c2 = o, and therefore A = Mb, B = Ma, C = M (a +b), so that the equation of the momental ellipsoid takes the form b2x2+a y2+(a2+b2) z1=s4.

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  • Yet, as she opened the car door, her gaze was drawn to the commuter plane.

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  • It is practically impossible to work with the sensitive film in contact with the reseau-film, not only because dust particles and contact would injure the silver film, but also because the plate-glass used for the photographic plates is seldom a perfect plane.

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  • The object glass of the micrometer-microscope is placed midway between the plane of the photographic plate and the plane of the micrometer webs.

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  • He also carried out many experiments in magneto-optics, and succeeded in showing, what Faraday had failed to detect, the rotation under the influence of magnetic force of the plane of polarization in certain gases and vapours.

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  • The principal trees are the oak, the valonia oak, the beech, ash, elm, plane, celtis, poplar and walnut, which give way in the higher regions to the pine and fir.

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  • Or the thallus may have a leaf-like form, the branches from the central threads which form the midrib growing out mainly in one plane and forming a lamina, extended right and left of the midrib.

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  • When a given initial cell of the cambium has once begun to produce cells of this sort it continues the process, so that a radial plate of parenchyma cells is formed stretching in one plane through the xylem and phloem.

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  • The division in all cases takes place by constriction, or by a simultaneous splitting along an equatorial plane.

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  • The angle which the earth's axis makes with the plane in which the planet revolves round the sun determines the varying seasonal distribution of solar radiation over the surface and the mathematical zones of climate.

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  • They are dualists, like the Bogomils, ascribing the body to a fall from a state when the soul was on the same plane as God.

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  • Two lines may be drawn from this point, one to each of the two rails, in a plane normal to the rails, and the ends of these lines, where they meet the rails, may be joined to complete a triangle, which may conveniently be regarded as a rigid frame resting on the rails.

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  • If therefore the outer rail is laid at a level above that of the inner rail at the curve, overturning will be resisted more than would be the case if both rails were in the same horizontal plane, since the tilting of the vehicle due to this " superelevation " diminishes the overturning moment, and also increases the restoring moment, by shortening in the one case and lengthening in the other the lever arms at which the respective forces act.

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  • In its simplest form, consisting of a ring fixed in the plane of the equator, the armilla is one of the most ancient of astronomical instruments.

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  • One possessed the power of turning the plane of the polarized ray to the right; the other possessed no rotary power.

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  • The first is a ventral flexure in the antero-posterior or sagittal plane; the result of this is to approximate the two ends of the alimentary canal.

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  • In a dextral Gastropod the shell is coiled in a right-handed spiral from apex to mouth, and the spiral also projects to the right of the median plane of the animal.

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  • In some forms the coiling disappears in the adult, leaving the shell simply conical as in Patellidae, Fissurellidae, &c., and in some cases the shell is coiled in one plane, e.g.

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  • Visceral sac and shell coiled in one plane; foot divided transversely into two parts, posterior part bearing an operculum, anterior part forming a fin provided with a sucker.

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  • The objects of religious knowledge are beyond the plane of history, or rather - in a thoroughly Gnostic and Neo-Platonic spirit - they are regarded as belonging to a supra-mundane history.

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  • When the foliation is undulose or sinuous the rocks are said to be crumpled, and have wavy splitting surfaces instead of nearly plane ones.

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  • It has small angle-windows to light the interior inclined plane or staircase, and is not broken into storeys with grouped windows as in the case of the Lombard bell-towers.

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  • The coil is so situated that, in its zero position when no current is passing through it, the plane of the coil is parallel to the direction of the lines of force of the field.

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  • The circle in which a sphere is cut by a plane is called a "great circle," when the cutting plane passes through the centre of sphere.

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  • Two elements define the position of the plane passing through the attracting centre in which the orbit lies.

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  • One of these is the position of the line MN through the sun at F in which the plane of the orbit cuts some fundamental plane of reference, commonly the ecliptic. This is called the line of nodes, and its position is specified by the angle which it makes with some fixed line FX in the fundamental plane.

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  • At one of the nodes, say N, the body passes from the south to the north side of the fundamental plane; this is called the ascending node.

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  • The other element is the inclination of the plane of the orbit to the fundamental plane, called the inclination simply.

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  • The angle from the pericentre to the actual radius vector, and the length of the latter being found, the angular distance of the planet from the node in the plane of the orbit is found by adding to the true anomaly the distance from the node to the pericentre.

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  • Planorbis has the spire of the shell in one plane.

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  • Arrangements connected with Claus' formula are obtained by placing six tetrahedra on the six triangles formed by the diagonals of a plane hexagon.

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  • It is thus seen that the ordinary plane representation of the structure of compounds possesses a higher significance than could have been suggested prior to crystallographical researches.

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  • Not until the third act does the great Wagner arbitrate in the struggle between amateurishness and theatricality in the music, though at all points his epoch-making stagecraft asserts itself with a force that tempts us to treat the whole work as if it were on the Wagnerian plane of Tannhauser's account of his pilgrimage in the third act.

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  • All objects on a map are required to be shown as projected horizontally upon a plane.

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  • It is obvious that the area of a group of mountains projected on a horizontal plane, such as is presented by a map, must differ widely from the area of the superficies or physical surface of those mountains exposed to the air.

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  • It is impossible to represent on a plane the whole of the earth's surface, or even a large extent of it, without a considerable amount of distortion.

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  • They are directed at first downwards by the side of the face, and then turn upwards and forwards, ending in the same plane as the eye.

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  • Ethically, too, the new doctrine stands on a higher plane, and represents, in its moral laws, a superior civilization.

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  • The great abstract ideas (considered directly and not merely in tacit use) which have dominated the science were due to them - namely, ratio, irrationality, continuity, the point, the straight line, the plane.

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  • His earliest publications, beginning with A Syllabus of Plane Algebraical Geometry (1860) and The Formulae of Plane Trigonometry (1861), were exclusively mathematical; but late in the year 1865 he published, under the pseudonym of "Lewis Carroll," Alice's Adventures in Wonderland, a work that was the outcome of his keen sympathy with the imagination of children and their sense of fun.

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  • They all crystallize in the monoclinic system, often, however, in forms closely resembling those of the rhombohedral or orthorhombic systems. Crystals have usually the form of hexagonal or rhomb-shaped scales, plates or prisms, with plane FIG.

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  • The plane of the optic axes may be either perpendicular or parallel to the plane of symmetry of the crystal, and according to its position two classes of mica are distinguished.

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  • To the first class, with the optic axial plane perpendicular to the plane of symmetry, belong muscovite, lepidolite, paragonite, and a rare variety of biotite called anomite; the second class includes zinnwaldite, phlogopite, lepidomelane and most biotites.

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  • For instance, those of a ternary form involve two classes which may be geometrically interpreted as point and line co-ordinates in a plane; those of a quaternary form involve three classes which may be geometrically interpreted as point, line and plane coordinates in space.

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  • Thus what have been called seminvariants are not all of them invariants for the general substitution, but are invariants for the particular substitution xl = X11 + J-s12, X 2 = 112 Again, in plane geometry, the most general equations of substitution which change from old axes inclined at w to new axes inclined at w' =13 - a, and inclined at angles a, l3 to the old axis of x, without change of origin, are x-sin(wa)X+sin(w -/3)Y sin w sin ' _sin ax y sin w a transformation of modulus sin w' sin w' The theory of invariants originated in the discussion, by George Boole, of this system so important in geometry.

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  • They are long and narrow; the sole is plane, but slopes from the fire-bridge towards the flue, so that the metal runs to the latter end to collect in pots placed outside the furnace.

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  • Christianity swept the whole discussion on to a higher plane.

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  • The combined mass of the earth and moon admits of being determined by its effect in changing the position of the plane of the orbit of Venus.

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  • The motion of the node of this plane is found with great exactness from observaMass, of the g tions of the transits of Venus.

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  • Upon one of these is based the principle of the mariner's compass, which is said to have been known to the Chinese as early as I ioo B.C., though it was not introduced into Europe until more than 2000 years later; a magnet supported so that its axis is free to turn in a horizontal plane will come to rest with its poles pointing approximately north and south.

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  • The compass needle is a little steel magnet balanced upon a pivot; one end of the needle, which always bears a distinguishing mark, points approximately, but not in general exactly, to the north,' the vertical plane through the direction of the needle being termed the magnetic meridian.

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  • For the practical observation of this phenomenon it is usual to employ a needle which can turn freely in the plane of the magnetic meridian upon a horizontal axis passing through the centre of gravity of the needle.

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  • The angle which the magnetic axis makes with the plane of the horizon is called the inclination or Along an irregular line encircling the earth in the neighbourhood of the geographical equator the needle takes up a horizontal position, and the dip is zero.

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  • The magnetic field due to a long straight wire in which a current of electricity is flowing is at every point at right angles to the plane passing through it and through the wire; its strength at any point distant r centimetres from the wire is H = 21/r, (2) i being the current in C.G.S.

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  • The equipotential surfaces are two series of ovoids surrounding the two poles respectively, and separated by a plane at zero potential passing perpendicularly through the middle of the axis.

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  • A magnet consisting of a series of plane shells of equal strength arranged at right angles to the direction of magnetization will be uniformly magnetized.

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  • The magnetic needle may be cemented horizontally across the back of a little plane or concave mirror, about or $ in.

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  • If a coil of insulated wire is suspended so that it is in stable equilibrium when its plane is parallel to the direction of a magnetic field, the transmission of a known electric current through the coil will cause it to be deflected through an angle which is a function of the field intensity.

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  • The intensity of a field may be measured by the rotation of the plane of polarization of light passing in the direction of the magnetic force through a transparent substance.

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  • It can be shown that if a current i circulates in a small plane circuit of area S, the magnetic action of the circuit for distant points is equivalent to that of a short magnet whose axis is perpendicular to the plane of the circuit and whose moment is iS, the direction of the magnetization being related to that of the circulating current as the thrust of a right-handed screw to its rotation.

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  • For those orbits whose projection upon a plane perpendicular to the field is righthanded, the period of revolution will be accelerated by the field (since the electron current is negative), and the magnetic moment consequently increased; for those which are left-handed, the period will be retarded and the moment diminished.

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  • Another was the magnetic rotation of the plane of polarization of light, which was effected in 1845, and for the first time established a relation between light and magnetism.

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  • Appendages of 2nd pair folding in a horizontal plane, completely chelate, the claw immovably united to the sixth segment.

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  • Appendages of 2nd pair folding in a vertical plane, not chelate, the claw long and movable.

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  • Laplace treated the subject from the point of view of the gradual aggregation and cooling of a mass of matter, and demonstrated that the form which such a mass would ultimately assume must be an ellipsoid of revolution whose equator was determined by the primitive plane of maximum areas.

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  • Archimedes' problem of dividing a sphere by a plane into two segments having a prescribed ratio,was first expressed as a cubic equation by Al Mahani, and the first solution was given by Abu Gafar al Hazin.

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  • We will now consider in detail the important case in which uniform plane waves are resolved at a surface coincident with a wave-front (OQ).

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  • We imagine a wave-front divided o x Q into elementary rings or zones - often named after Huygens, but better after Fresnelby spheres described round P (the point at which the aggregate effect is to be estimated), the first sphere, touching the plane at 0, with a radius equal to PO, and the succeeding spheres with radii increasing at each step by IX.

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  • When the primary wave is plane, the area of the first Fresnel zone is 7rXr, and, since the secondary waves vary as r 1, the intensity is independent of r, as of course it should be.

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  • An interesting exception to the general rule that full brightness requires the existence of the first zone occurs when the obstacle assumes the form of a small circular disk parallel to the plane of the incident waves.

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  • The amplitude of the light at any point in the axis, when plane waves are incident perpendicularly upon an annular aperture, is, as above, cos k(at-r 1)-cos k(at-r 2) =2 sin kat sin k(r1-r2), r2, r i being the distances of the outer and inner boundaries from the point in question.

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  • Taking co-ordinates in the plane of the screen with the centre of the wave as origin, let us represent M by, n, and P (where dS is situated) by x, y, z.

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  • The incident waves are thus plane, and are limited to a plane aperture coincident with a wave-front.

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  • What is seen through the eye-piece in any case is the same as would be depicted upon a screen in the focal plane.

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  • If the aperture be increased, not only is the total brightness over the focal plane increased with it, but there is also a concentration of the diffraction pattern.

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  • It has already been suggested that the principle of energy requires that the general expression for I 2 in (2) when integrated over the whole of the plane, n should be equal to A, where A is the area of the aperture.

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  • In these expressions we are to replace p by ks/f, or rather, since the diffraction pattern is symmetrical, by kr/f, where r is the distance of any point in the focal plane from the centre of the system.

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  • Let AoBo be a plane wave-surface of the light before it falls upon the prisms, AB the corresponding wave-surface for a particular part of the spectrum after the light has passed the prisms, or after it has passed the eye-piece of the observing telescope.

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  • If the source be a point or a line, and a collimating lens be used, the incident waves may be regarded as plane.

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  • It may be worth while to examine further the other variations from correct ruling which correspond to the various terms expressing the deviation of the wave-surface from a perfect plane.

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  • When the plane zx is not a plane of symmetry, we have to consider the terms in xy, 2 y, and y 3 .

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  • He has also shown how to rule a plane surface with lines so disposed that the grating shall of itself give well-focused spectra.

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  • If we consider for the present only the primary plane of symmetry, the figure is reduced to two dimensions.

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  • This is the ordinary formula for a reflecting plane grating, and it shows that the spectra are formed in the usual directions.

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  • It may be remarked that these calculations apply to the rays in the primary plane only.

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  • In Rowland's dividing engine the screws were prepared by a special process devised by him, and the resulting gratings, plane and concave, have supplied the means for much of the best modern optical work.

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  • In theoretical investigations these problems are usually treated as of two dimensions only, everything being referred to the plane passing through the luminous point and perpendicular to the diffracting edges, supposed to be straight and parallel.

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  • When, in order to apply Huygens's principle, the wave is supposed to be broken up, the phase is the same at every element of the surface of resolution which lies upon a line perpendicular to the plane of reference, and thus the effect of the whole line, or rather infinitesimal strip, is related in a constant manner to that of the element which lies O in the plane of reference, and may be considered to be represented thereby.

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  • These equations simplify very much in their application to plane waves.

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  • Then the displacement at 0 will take place in a direction perpendicular to 0 1 0, and lying in the plane Z0 1 0; and, if 1' be the displacement at 0, reckoned positive in the direction nearest to that in which the incident vibrations are reckoned positive, = 4?y (1 +cos 0) sin 4 f' (bt - r).

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  • Imagine a flexible lamina to be introduced so as to coincide with the plane at which resolution is to be effected.

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  • The conception of the lamina leads immediately to two schemes, according to which a primary wave may be supposed to be broken up. In the first of these the element dS, the effect of which is to be estimated, is supposed to execute its actual motion, while every other element of the plane lamina is maintained at rest.

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  • When the secondary disturbance thus obtained is integrated with respect to dS over the entire plane of the lamina, the result is necessarily the same as would have been obtained had the primary wave been supposed to pass on without resolution, for this is precisely the motion generated when every element of the lamina vibrates with a common motion, equal to that attributed to dS.

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  • The intensity of light is, however, more usually expressed in terms of the actual displacement in the plane of the wave.

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  • This displacement, which we may denote by; is in the plane containing z and r, and perpendicular to the latter.

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  • We will now apply (18) to the investigation of a law of secondary disturbance, when a primary wave = sin (nt - kx) (19) is supposed to be broken up in passing the plane x = o.

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  • The force operative upon the positive half is parallel to OZ, and of amount per unit of area equal to - b 2 D = b 2 kD cos nt; and to this force acting over the whole of the plane the actual motion on the positive side may be conceived to be due.

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  • Thus, to refer again to the acoustical analogue in which plane waves are incident upon a perforated rigid screen, the circumstances of the case are best represented by the first method of resolution, leading to symmetrical secondary waves, in which the normal motion is supposed to be zero over the unperforated parts.

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  • Of the total amount of light falling on such a sphere, part is reflected or scattered at the incident surface, so rendering the drop visible, while a part will enter the drop. Confining our attention to a ray entering in a principal plane, we will determine its deviation, i.e.

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  • The nucleolus is elongated, and its longest measurement lies in the direction of the equatorial plane of the nucleus.

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  • Generally, however, Hahnemann's views contradict those of Brown, though moving somewhat in the same plane.

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  • The minimum grade is that which will enable the loaded cars in travelling down the plane to pull up the empty cars.

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  • At the head of the plane is mounted a drum or sheave, and around it passes a rope, one end of which is attached to the loaded cars at the top, the other to the empty cars at the foot.

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  • An engine plane is an inclined road, up which loaded cars are hauled by a stationary engine and rope, the empty cars running down by gravity, dragging the rope after them.

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  • These books, except the Definitiones, mostly consist of directions for obtaining, from given parts, the areas or volumes, and other parts, of plane or solid figures.

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  • For the purpose of rendering this minute examination possible, opposite plane surfaces of the glass are ground approximately flat and polished, the faces to be polished being so chosen as to allow of a view through the greatest possible thickness of glass; thus in slabs the narrow edges are polished.

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  • The first step in this process is that of grinding the surface down until all projections are removed and a close approximation to a perfect plane is obtained.

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  • When the surface of the glass has been ground down to a plane, the surface itself is still " grey," i.e.

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  • This term is applied to blown sheet-glass, whose surface has been rendered plane and brilliant by a process of grinding and polishing.

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  • Dubuat considered that if water were a perfect fluid, and the channels in which it flowed infinitely smooth, its motion would be continually accelerated, like that of bodies descending in an inclined plane.

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  • A fluid is a substance which yields continually to the slightest tangential stress in its interior; that is, it can be divided very easily along any plane (given plenty of time if the fluid is viscous).

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  • It follows that when the fluid has come to rest, the tangential stress in any plane in its interior must vanish, and the stress must be entirely normal to the plane.

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  • The pressure at any point cf a plane in the interior of a fluid is the intensity of the normal thrust estimated per unit area of the plane.

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  • Thus, if a thrust OP lb acts on a small plane area DA ft.

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  • Take any two arbitrary directions in the plane of the paper, and draw a small isosceles triangle abc, whose sides are perpendicular to the two directions, and consider the equilibrium of a small triangular prism of fluid, of which the triangle is the cross section.

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  • As gravity and the fluid pressure on the sides of the prism act at right angles to AB, the equilibrium requires the equality of thrust on the ends A and B; and as the areas are equal, the pressure must be equal at A and B; and so the pressure is the same at all points in the same horizontal plane.

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  • If the fluid is a liquid, it can have a free surface without diffusing itself, as a gas would; and this free surface, being a surface of zero pressure, or more generally of uniform atmospheric pressure, will also be a surface of equal pressure, and therefore a horizontal plane.

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  • The land has hills and valleys, but the surface of water at rest is a horizontal plane; and if disturbed the surface moves in waves.

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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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  • Thus if the plane is normal to Or, the resultant thrust R =f fpdxdy, (r) and the co-ordinates x, y of the C.P. are given by xR = f f xpdxdy, yR = f f ypdxdy.

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  • This shows that the C.P. is the antipole of the line of intersection of its plane with the free surface with respect to the momental ellipse at the C.G.

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  • Suppose the ship turns about an axis through F in the water-line area, perpendicular to the plane of the paper; denoting by y the distance of an element dA if the water-line area from the axis of rotation, the change of displacement is EydA tan 8, so that there is no change of displacement if EydA = o, that is, if the axis passes through the C.G.

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  • I n a straight uniform current of fluid of density p, flowing with velocity q, the flow in units of mass per second across a plane area A, placed in the current with the normal of the plane making an angle 0 with the velocity, is oAq cos 0, the product of the density p, the area A, and q cos 0 the component velocity normal to the plane.

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  • Taking the axis of x for an instant in the normal through a point on the surface H = constant, this makes u = o, = o; and in steady motion the equations reduce to dH/dv=2q-2wn = 2gco sin e, (4) where B is the angle between the stream line and vortex line; and this holds for their projection on any plane to which dv is drawn perpendicular.

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  • For a plane boundary the image is the optical reflection of the vortex.

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  • For example, a pair of equal opposite vortices, moving on a line parallel to a plane boundary, will have a corresponding pair of images, forming a rectangle of vortices, and the path of a vortex will be the Cotes' spiral r sin 20 = 2a, or x-2+y-2=a-2; (io) this is therefore the path of a single vortex in a right-angled corner; and generally.

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  • Negative values of n must be interpreted by a streaming motion on a parallel plane at a level slightly different, as on a double Riemann sheet, the stream passing from one sheet to the other across a cut SS' joining the foci S, S'.

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  • The resultant hydrostatic thrust across any diametral plane of the cylinder will be modified, but the only term in the loss of head which exerts a resultant thrust on the whole cylinder is 2mU sin Olga, and its thrust is 27rpmU absolute units in the direction Cy, to be counteracted by a support at the centre C; the liquid is streaming past r=a with velocity U reversed, and the cylinder is surrounded by a vortex.

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  • 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.

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  • Projected perpendicularly against a plane boundary, the motion is determined by an equal opposite vortex ring, the optical image; the vortex ring spreads out and moves more slowly as it approaches the wall; at the same time the molecular rotation, inversely as the cross-section of the vortex, is seen to increase.

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  • By analogy with the spin of a rigid body, the component spin of the fluid in any plane at a point is defined as the circulation round a small area in the plane enclosing the point, divided by twice the area.

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  • 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.

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  • To determine the motion of a jet which issues from a vessel with plane walls, the vector I must be constructed so as to have a constant (to) (II) the liquid (15) 2, integrals;, (29) (30) (I) direction 0 along a plane boundary, and to give a constant skin velocity over the surface of a jet, where the pressure is constant.

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  • Over the ellipsoid, p denoting the length of the perpendicular from the centre on a tangent plane, px _ pv _ _ pz 1= a2+X' b +A' n c2+A p2x2 + p2y2 p2z2 I (a2 - + X)2 (b 2 +x)2 + (0+X)2, p 2 = (a2+A)12+(b2+X)m2+(c2+X)n2, = a 2 1 2 +b 2 m 2 +c 2 n 2 +X, 2p d = ds; (8) Thence d?

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  • The continuity is secured if the liquid between two ellipsoids X and X 11 moving with the velocity U and 15 1 of equation (II), is squeezed out or sucked in across the plane x=o at a rate equal to the integral flow of the velocity I across the annular area a l.

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  • A card will show the influence of the couple N if projected with a spin in its plane, when it will be found to change its aspect in the air.

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  • The plane projection of molecular structures which differ stereochemically is discussed under Stereoisomerism; in this place it suffices to say that, since the terminal groups of the hexaldose molecule are different and four asymmetric carbon atoms are present, sixteen hexaldoses are possible; and for the hexahydric alcohols which they yield on reduction, and the tetrahydric dicarboxylic acids which they give on oxidation, only ten forms are possible.

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  • The orthogonal projection of a section of this surface by a plane containing one of the perpendiculars and inclined to the axis is the quadratrix.

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  • Any plane section of the screw (plectoidal of Pappus) surface so obtained is the quadratrix.

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  • This disk is carried on an axle inclined to the line of draught, and also to a vertical plane.

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  • His arrangement of concave and plane mirrors, by which the realistic images of objects inside the house or in the street could be rendered visible though intangible, there alluded to, may apply to a camera on Cardan's principle or to a method of aerial projection by means of concave mirrors, which Bacon was quite familiar with, and indeed was known long before his time.

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  • In the Diversarum Speculationum Mathematicarum et Physicarum (1585), by the Venetian Giovanni Battista Benedetti, there is a letter in which he discusses the simple camera obscura and mentions the improvement some one had made in it by the use of a double convex lens in the aperture; he also says that the images could be made erect by reflection from any plane mirror.

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  • The great variety in the apparent motions of meteors proves that they are not directed from the plane of the ecliptic; hence their orbits are not like the orbits of planets and short-period comets, which are little inclined, but like the orbits of parabolic comets, which often have great inclinations.

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  • In this style the surface of the design is not raised above the general plane of the field, but an effect of projection is obtained either by recessing the whole space immediately surrounding the design, or by enclosing the latter in a scarped frame.

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  • All this work was in the style known as hira-makie (flat decoration); that is to say, having the decorative design in the same plane as the ground.

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  • In fact the uniformity of brass and bell-metal is only superficial; if we adopt the methods described in the article Metallography, and if, after polishing a plane face on a bit of gun-metal, we etch away the surface layer and examine the new surface with a lens or a microscope, we find a complex pattern of at least two materials.

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  • Only once, and for a brief period, in the reigns of David and Solomon did the Hebrews rise to anything like an equal plane of political importance with their immediate neighbours.

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  • The buildings are now in ruins, but the view from the pavilions, shaded by splendid plane trees on the terraced gardens formed on the slope of the mountain, is said to be very beautiful.

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  • It is desirable for two reasons that the image should lie in the plane of the paper, and this can be secured by placing a suitable lens between the object and the prism.

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  • If the image does not lie in the plane of the paper, it is impossible to see it and the pencil-point clearly at the same time.

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  • Exceptions occur in the case of the satellites of Uranus, which are nearly perpendicular to the plane of the orbit.

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  • One-half of this plane remains fixed, the other half is movable.

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  • When the inclination of the movable half with respect to the axis of the telescope is changed by rotation about an axis at right angles to the plane of division, two images are produced.

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  • The tail rope, which is of lighter section than the main one, is coiled on the second drum, passes over similar guide sheaves placed near the roof or side of the gallery round a pulley at the bottom of the plane, and is fixed to the end of the train or set of tubs.

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  • The chain passes over a pulley driven by the engine, placed at such a height as to allow it to rest upon the tops of the tubs, and round a similar pulley at the far end of the plane.

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  • The moments of the components of these actions and reactions in a plane to which the axis of rotation is at right angles are the two aspects of the torque acting, and therefore the torque acting on B through the shaft is measured by the torque required to hold A still.

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  • The contradiction can only be suppressed if the ego itself opposes to itself the non-ego, places it as an Anstoss or plane on which its own activity breaks and from which it is reflected.

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  • Between the rising swells of long-leaf pine lands are impenetrable thickets of hawthorn, holly, privet, plane trees and magnolias.

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  • The matter is the sensible thing which in accordance with Christ's institution can be raised to a sacramental plane.

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  • Owing to the conical shape of the early muzzle-loading guns, if one trunnion were higher than the other, the " line of metal " would no longer be in the same vertical plane as the axis; in consequence of this, if a gun with, say, one wheel higher than the other were layed by this line, the axis would point off the target to the side of the lower wheel.

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  • In coast defence artillery, owing to the fact that the guns are on fixed mountings at a constant height (except for rise and fall of tide) above the horizontal plane on which their targets move, and that consequently the angle of sight and quadrant elevation for every range can be calculated, developments in sights, in a measure, gave way to improved means of giving quadrant elevation.

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  • One great drawback to this system was that elevation was given with reference to the plane of the racers upon which the mounting moved, and as this was not always truly horizontal grave errors were introduced.

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  • 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.

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  • In the addax (Addax nasomaculatus), which is a distinct species common to North Africa and Syria, the ringed horns form an open spiral ascending in the plane of the face, and there is long, shaggy, dark hair on the fore-quarters in winter.

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  • The various species of oryx differ from Hippotragus by the absence of the white eye-tufts, and by the horns sloping backwards in the plane of the face.

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  • Aristarchus is also said to have invented two sun-dials, one hemi spherical, the so-called scaphion, the other plane.

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  • If a .JP solid circle be fixed in any one position and a tube be pivoted on its centre so as to move; and if the line C D be drawn upon the circle pointing towards any object Q in the heavens which lies in the plane of the circle, by turn ing the tube A B towards any other object P in the plane of the circle, the angle B 0 D will be the angle subtended by the two objects P and Q at the eye.

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  • Even in the second sense, the term is a very wide one, since it comprises the measurement of angles (plane and solid), lengths, areas and volumes.

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  • The first group comprises such subjects as land-surveying; here the measurements in the elementary stages take place in a plane, and the consideration of volumes necessarily constitutes a later stage; and the figures to be measured are mostly not movable, so that triangulation plays an important part.

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  • The second group comprises the mechanic arts, in which the bodies to be measured are solid bodies which can be handled; in these cases plane figures appear mainly as sections of a solid.

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  • The next stage is geometrical mensuration, where geometrical methods are applied to determine the areas of plane rectilinear figures and the volumes of solids with plane faces.

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  • In the case of plane figures, the congruence is tested by an imaginary superposition of one figure on the other; but this may more simply be regarded as the superposition, on either figure, of the image of the other figure on a contiguous plane.

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  • The rectangle, for instance, has so far been regarded as a plane figure bounded by one pair of parallel straight lines and another pair at right angles to them, so that the conception of " rectangularity " has had reference to boundary rather than to content; analytically, the rectangle must be regarded as the figure generated by an ordinate of constant length moving parallel to itself with one extremity on a straight line perpendicular to it.

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  • This is the simplest case of generation of a plane figure by a moving ordinate; the corresponding figure for generation by rotation of a radius vector is a circle.

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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 ordinary definition of a circle is equivalent to definition as the figure generated by the rotation of a radius of constant length in a plane, and is thus essentially analytical.

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  • This implies the treatment of a plane or solid figure as being wholly comprised between two parallel lines or planes, regarded by convention as being vertical; the figure being generated by an ordinate or section moving at right angles to itself through a distance which is called the breadth of the figure.

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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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  • Two adjoining faces in the same plane may together make a trapezium.

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  • If R and S are the ends of a prismoid, A and B their areas, h the perpendicular distance between them, and C the area of a section by a plane parallel to R and S and midway between them, the volume of the prismoid is *h(A+4C+B).

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  • The moment of a figure with regard to a plane is found by dividing the figure into elements of volume, area or length, multiplying each element by its distance from the plane, and adding the products.

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  • In the case of a plane area or a plane continuous line the moment with regard to a straight line in the plane is the same as the moment with regard to a perpendicular plane through this line; it is the sum of the products of each element of area or length by its distance from the straight line.

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  • The centroid of a figure is a point fixed with regard to the figure, and such that its moment with regard to any plane (or, in the case of a plane area or line, with regard to any line in the plane) is the same as if the whole volume, area or length were concentrated at this point.

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  • We sometimes require the moments with regard to a line or plane through the centroid.

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  • These formulae also hold for converting moments of a solid figure with regard to a plane into moments with regard to a parallel plane through the centroid; x being the distance between the two planes.

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  • A line through the centroid of a plane figure (drawn in the plane of the figure) is a central line, and a plane through the centroid of a solid figure is a central plane, of the figure.

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  • The first moment of a plane figure with regard to a line in its plane may be regarded as obtained by dividing the area into elementary strips by a series of parallel lines indefinitely close together, and concentrating the area of each strip at its centre.

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  • This also holds for higher moments, provided that the edges of the elementary strips or prisms are parallel to the line or plane with regard to which the moments are taken.

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  • Any plane figure might be converted into an equivalent trapezette by an extension of the method of § 25 (iv).

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  • The corresponding solid figure, in its most general form, is such as would be constructed to represent the relation of a magnitude E to two magnitudes F and G of which it is a function; it would stand on a plane base, and be comprised within a cylindrical boundary whose cross-section might be of any shape.

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  • A plane parallel to either pair of sides of the briquette is a " principal plane."

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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 section by any intermediate parallel plane will be called a " cross-section."

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  • If the area of the cross-section, in every position, is known in terms of its distance from one of the bounding planes, or from a fixed plane A parallel to them, the volume of the solid can be expressed in terms of the area of a trapezette.

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  • The volume of a briquette can be found in this way if the area of the section by any principal plane can be expressed in terms of the distance of this plane from a fixed plane of the same set.

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  • It follows from §§ 48 and 51 that, if V is a solid figure extending from a plane K to a parallel plane L, and if the area of every cross-section parallel to these planes is a quadratic function of the distance of the section from a fixed plane parallel to them, Simpson's formula may be applied to find the volume of the solid.

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  • In the case of a pyramid, of height h, the area of the section by a plane parallel to the base and at distance x from the vertex is clearly x 2 /h 2 X area of base.

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  • For a tetrahedron, two of whose opposite edges are AB and CD, we require the area of the section by a plane parallel to AB and CD.

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  • Let the distance between the parallel planes through AB and CD be h, and let a plane at distance x from the plane through AB cut the edges AC, up -f- .

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  • By drawing Ac and Ad parallel to BC and BD, so as to meet the plane through CD in c and d, and producing QP and RS to meet Ac and Ad in q and r, we see that the area of Pqrs is (x/h - x 2 /h 2) X area of cCDd; this also is a quadratic function of x.

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  • To extend these methods to a briquette, where the ordinate u is an algebraical function of x and y, the axes of x and of y being parallel to the sides of the base, we consider that the area of a section at distance x from the plane x = o is expressed in terms of the ordinates in which it intersects the series of planes, parallel to y=o, through the given ordinates of the briquette (§ 44); and that the area of the section is then represented by the ordinate of a trapezette.

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  • The areas of the sides for which 0 and x=xo+2h, and of the section by the plane x=xo+h, may be found by Simpson's second formula; call these Ao and A2, and Al.

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  • The area of the section by a plane at distance x from the edge 0 is a function of x whose degree is the same as that of u.

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  • The methods of §§ 59 and 60 can similarly be extended to finding the position of the central ordinate of a briquette, or the mean q th of elements of the briquette from a principal plane.

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  • The plane of the joint orbit, in which no deviation from circularity has yet been detected, nearly coincides with the line of sight.

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  • If we take one of these spheres a distance from the source very great as compared with a single wave-length, and draw a radius to a point on the sphere, then for some little way round that point the sphere may be regarded as a plane perpendicular to the radius or the line of propagation.

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  • Every particle in the plane will have the same displacement and the same velocity, and these will be perpendicular to the plane and parallel to the line of propagation.

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  • The waves for some little distance on each side of the plane will be practically of the same size.

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  • In fact, we may neglect the divergence, and may regard them as " plane waves."

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  • Experiments may be made with plane and curved mirrors to verify these laws, but it is necessary to use short waves, in order to diminish diffraction effects.

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  • At the instant that the original wave reaches F the wave from E has travelled to a circle of radius very nearly equal to EF-not quite, as S is not quite in the plane of the rails.

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  • This second plate is capable of rotation about an axis perpendicular to its plane and passing through its centre.

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  • On one prong of each fork is fixed a small plane mirror.

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  • The two forks are fixed so that one vibrates in a vertical, and the other in a horizontal, plane, and they are so placed that a converging beam of light received on one mirror is reflected to the other and then brought to a point on a screen.

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  • Imagine now that a fork with black prongs is held near the cylinder with its prongs vertical and the plane of vibration parallel to.

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  • An observer in the plane of the motion can easily hear a change in the pitch as the pitch-pipe moves to and from him.

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  • In a common form of electrically maintained fork, the fork is set horizontal with its prongs in a vertical plane, and a small electro-magnet is fixed between them.

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  • An "anchor ring" or "tore" results when a circle revolves about an axis in its plane.

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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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  • Two spheres intersect in a plane, and the equation to a system of spheres which intersect in a common circle is x 2 + y 2 + z 2 +2Ax -fD = o, in which A varies from sphere to sphere, and D is constant for all the spheres, the plane yz being the plane of intersection, and the axis of x the line of centres.

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  • In mathematical geography the problem of representing the surface of a sphere on a plane is of fundamental importance; this subject is treated in the article MAP.

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  • The opening bridge between the river towers consists of two leaves or bascules, pivoted near the faces of the piers and rotating in a vertical plane.

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  • They are quite distinct from the somewhat similar orders of "virgins" and "widows," who belonged to a lower plane in the ecclesiastical system.

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  • A train of ideas which strongly impressed itself on Clerk Maxwell's mind, in the early stages of his theoretical views, was put forward by Lord Kelvin in 1858; he showed that the special characteristics of the rotation of the plane of polarization, discovered by Faraday in light propagated along a magnetic field, viz.

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  • Those who did not adopt the monastic life endeavoured on a lower plane and in a less perfect way to realize the common ideal, and by means of penance to atone for the deficiencies in their performance.

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  • Later still he engaged in the study of the relations between chemical constitution and rotation of the plane of polarization in a magnetic field, and enunciated a law expressing the variation of such rotation in bodies belonging to homologous series.

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  • The best known of these, which is called Legendre's theorem, is usually given in treatises on spherical trigonometry; by means of it a small spherical triangle may be treated as a plane triangle, certain corrections being applied to the angles.

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  • To become a medusa, the actinula grows scarcely at all in the direction of the principal axis, but greatly along a plane at right angles to it.

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  • So also the angle /3 must be increased by S to obtain the angle at which the shot strikes a horizontal plane - the water, for instance.

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  • Near the town was the famous fountain of Sauros, inclosed by fruit-bearing poplars; and not far from this was another spring, overhung by an evergreen plane tree which in popular belief marked the scene of the amours of Zeus and Europa.

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  • The struggle for freedom called forth a deeper sense of the unity of the people of the one Yahweh, and in so doing raised religion to a loftier plane; for a faith which unites a nation is necessarily a higher moral force than one which only unites a township or a.

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  • A secondary standard measure for dry goods is the bushel of 1824, containing 8 imperial gallons, represented by a hollow bronze cylinder having a plane base, its internal diameter bring double its depth.

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  • Flournoy has shown that these utterances may reach a higher plane and form a real language, which is, however, based on one already known to the speaker.

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  • The auxiliary magnet has a plane mirror attached, the plane of which is at right angles to the axis of the magnet.

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  • Others, which may be called " earth-tiltings," show themselves by a slow bending and unbending of the surface, so that a post stuck in the ground, vertical to begin with, does not remain vertical, but inclines now to one side and now to another, the plane of the ground in which it stands shifting relatively to the horizon.

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  • As this position is approached the period of swing becomes greater and greater, and sensibility to slight tilting at right angles to the plane of o'o"m is increased.

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  • Of the small family of the Tilopteridaceae our knowledge is as yet inadequate, but they probably present the only case of pronounced oogamy among Phaeosporeae.;.They are filamentous forms, exhibiting, however, a tendency to division in more than one plane, even in the vegetative parts.

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  • There are other forms of shaft kiln, such as the Schneider, in which there is a burning zone, a heating and cooling zone as in the Dietzsch, but no horizontal stage, the whole shaft being in the same vertical plane.

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  • It may be defined as a section of a right circular cone by a plane parallel to a tangent plane to the cone, or as the locus of a point which moves .so that its distances from a fixed point and a fixed line are equal.

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

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  • Sycamore (Ater pseudo-platanus), sometimes mistakenly called the plane tree, is common in Germany and Britain and in the eastern states of North America.

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  • One of the simplest consists of a plane mirror rigidly connected with a revolving axis so that the angle be tween the normal to the mirror and the axis of the instrument equals half the sun's polar distance, the mirror being adjusted so that the normal has the same right ascension as the sun.

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  • By adjusting the right ascension of the plane ABC and rotating the axis with the angular velocity of the sun, it follows that BC will be the direction of the solar rays throughout the day.

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  • X is the mirror rotating about the point E, and placed so that (if EB is the horizontal direction in which the rays are to be reflected) (I) the normal CE to the mirror is jointed to BC at C and is equal in length to BE, (2) the rod DBC passes through a slot in a rod ED fixed to, and in the plane of, the mirror.

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  • For his demonstration in 1851 of the diurnal motion of the earth by the rotation of the plane of oscillation of a freely suspended, long and heavy pendulum exhibited by him at the Pantheon in Paris, and again in the following year by means of his invention the gyroscope, he received the Copley medal of the Royal Society in 1855, and in the same year he was made physical assistant in the imperial observatory at Paris.

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  • Nearly all patterns are the developments of the envelopes of geometrical solids of regular or irregular outlines, few of plane faces; when they are made up of combinations of plane faces, or of faces curved in one plane only, there is no difference in dealing with thin sheets or thick plates.

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  • All the works in sheet metal that are bent in one plane only are easily made.

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  • It contains an old summer palace, overshadowed by plane trees, with numerous springs, and a fine mosque and shrine.

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  • To fix a weighted point and a weighted plane in Euclidean space we require 8 scalars, and not the 12 scalars of a tri-quaternion.

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  • Vq/Sq, and that of the foot of perpendicular from centre on plane is Srg i.

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  • The axis of the member xQ+x'Q' of the second-order complex Q, Q' (where Q=nq+wr, Q'=nq'+wr' and x, x' are scalars) is parallel to a fixed plane and intersects a fixed transversal, viz.

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  • Both mirrors are usually concave; if plane, a concave lens is placed immediately before them.

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  • On the west coast the ilex, plane, oak, valonia oak, and pine predominate.

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  • K need not be confined to one plane.

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  • As a simple example of the geometrical method of treating statical problems we may consider the equilibrium of a particle on a rough inclined plane.

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  • The relations between this force P, the gravity W of the body, and the reaction S of the plane are then determined by a triangle of forces HKL.

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  • Thus a fdrce can be uniquely resolved into two components acting in two assigned directions in the same plane with it by an inversion of the parallelogram construction of fig.

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  • Plane Kinematics of a Rigid Body.The ideal rigid body, is one in which the distance between any two points is invariable.

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  • The position of a lamina movable in its own plane is determinate when we know the positions of any two points A, B of it.

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  • The lamina when perfectly free to move in its own plane is said to have three degrees of freedom.

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  • We now restrict ourselves for the present to the systems of forces in one plane.

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  • The sum of the moments of the two forces of a couple is the same about any point in the plane.

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  • A system of forces represented completely by the sides of I plane polygon taken in order is equivalent to a couple whosc moment is represented by twice the area of the polygon; this is proved by taking moments about any point.

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  • The points thus obtained are evidently the vertices of a polyhedron with plane faces.

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  • Two plane figures so related are called reciprocal, since the properties of the first figure in relation to the second are the same as those of the second with respect to the first.

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  • If we project both polyhedra orthogonally on a plane perpendicular to the axis of the paraboloid, we obtain two figures which are reciprocal, except that corresponding lines are orthogonal instead of parallel.

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  • When a plane frame which is just rigid is subject to a given system of equilibrating extraneous forces (in its own plane) acting on the joints, the stresses in the bars are in general uniquely determinate.

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  • A plane frame which can be built up from a single bar by suc cessive steps, at each of which a new joint is introduced by tw new bars meeting there, is called a simple frame; it is obviously just rigid.

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  • For instance, the position of a theodolite is fixed by the fact that its rounded feet rest in contact with six given plane surfaces.

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  • The composition of finite rotations about parallel axes is, a particular case of the preceding; the radius of the sphere is now infinite, and the triangles are plane.

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  • If AB, AC represent infinitesimal rotations about intersecting axes, the consequent displacement of any point 0 in the plane BAC will be at right angles to this plane, and will be represented by twice the sum of the areas OAB, OAC, taken with proper signs.

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  • In particular cases the cylindroid may degenerate into a plane, the pitches being then all equal.

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  • Hence any three-dimensional system can be reduced to a single force R acting in a certain line, together with a couple G in a plane perpendicular to the line.

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  • From the analogy of couples to translations which was pointed out in 7, we may infer that a couple is sufficiently represented by a free (or non-localized) vector perpendicular to its plane.

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  • Thus, let the plane of the paper be perpendicular to the planes of two couples, and therefore perpendicular to the line of intersection of these planes.

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  • The two forces at B will cancel, and we are left with a couple of moment P.AC in the plane AC. If we draw three vectors to represent these three couples, they will be perpendicular and proportional to the respective sides of the triangle ABC; hence the third vector is the geometric sum of the other two.

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  • By properly choosing 0 we can make the plane of the couple perpendicular to the resultant force.

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  • Since the given wrench can be replaced by a force acting through any assigned point P, and a couple, the locus of the null-lines through P is a plane, viz, a plane perpendicular to the vector which represents the couple.

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  • The plane in question is called the null-plane of P. If the null-plane of P pass through Q, the null-plane of will pass through P, since PQ is a null-line.

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  • Again, any plane w is the locus of a system of null-lines meeting in a point, called the null-point of c. If a plane revolve about a fixed straight line p in it, its ntill-point describes another straight line p, which is called the conjugate line of p. We have seen that the wrench may be replaced by two forces, one of which may act in any arbitrary line p. It is now evident that the second force must act in the conjugate line p, since every line meeting p, p is a null-line.

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  • If we take any polyhedron with plane faces, the null-planes of its vertices with respect to a given wrench will form another polyhedron, and the edges of the latter will be conjugate (in the above sense) to those of the former.

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  • Projecting orthogonally on a plane perpendicular to the central axis we obtain two reciprocal figures.

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  • It is to be noticed that All need not be in the same plane with AB, AC.

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  • The work of a couple in any infinitely small rotation of a rigid body about an axis perpendicular to the plane of the couple is equal to the product of the moment of the couple into the angle of rotation, proper conventions as to sign being observed.

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

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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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  • We proceed to the theory of the plane, axial and polar quadratic moments of the system.

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  • Evidently the quadratic moment for a variable plane through 0 will have a stationary value when, and only when, the plane coincides with a principal plane of (26).

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  • Now consider the tangent plane w at any point P of a confocal, the tangent plane fii at an adjacent point N, and a plane of through P parallel to of.

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  • The directions of these axes are determined by the property (24), and therefore coincide with those of the principal axes of inertia at 0, as already defined in connection with the theory of plane quadratic moments.

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  • The graphical methods of determining the moment of inertia of a plane system of particles with respect to any line in its plane may be briefly noticed.

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  • The small oscillations of a simple pendulum in a vertical plane also come under equation (5).

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  • For example, the path of a particle projected anyhow under gravity will obviously be confined to the vertical plane through the initial direction of motion.

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  • The range on a horizontal plane through 0 is got by putting y=o, viz, it is 2uovo!g.

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  • The motion wiLl evidently be in one plane, which we take as the plane z=o.

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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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  • Hodograph.The motion of a particle subject to a force which passes always through a fixed point 0 is necessarily in a plane orbit.

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  • A plane through G perpendicular to this vector has a fixed direction in space, and is called the invariable plane; it may sometimes be conveniently used as a plane of reference.

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  • A pendulum is constructed with two parallel knife-edges as nearly as possible in the same plane with G, the position of one of them being adjustable.

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  • If the axis of x be taken parallel to the slope of the plane, with x increasing downwards, we have -

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  • The axis of resultant angular momentum is therefore normal to the tangent plane at J, and does not coincide with OJ unless the latter be a principal axis.

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  • Take, for example, the case of a sphere rolling on a plane; and let the axes Ox, Oy be drawn through the centre parallel to the plane, so that the equation of the latter is 1=cf.

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  • Again, since the point of the sphere which is in contact with the plane is instantaneously at rest, we have the geometrical relations u+qa=0, v+pa=o, W0, (20) by (12).

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  • The acceleration of the centre is therefore the same as if the plane were smooth and the mass of the sphere were increased by C/a.

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  • Thus the centre of a sphere rolling under gravity on a plane of inclination a describes a parabola with an acceleration g sin a/(I+C/Ma)

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  • Since w varies as p, it follows that OH is constant, and the tangent plane at J is therefore fixed in space.

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  • The motion of the body relative to 0 is therefore completely represented if we imagine the momental ellipsoid at 0 to roll without sliding on a plane fixed in space, with an angular velocity proportional at each instant to the radius-vector of the point of contact.

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  • The fixed plane is parallel to the invariable plane at 0, and the line OH is called the invariable line.

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  • The trace of the point of contact J on the fixed plane is the herpolhode.

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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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  • If this couple be absent, the axis will be tilted out of the horizontal plane in such a sense that the direction of the spin n approxi mates to that of the azimuthal rota- K

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  • Again, take the case of a circular disk rolling in steady motion on a horizontal plane.

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  • The ratio of the axes of the ellipse is sec a, the longer axis being in the plane of 0.

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  • Let r be the distance of a point P from a fixed origin 0, 0 the angle which OP makes with a fixed direction OZ, il the azimuth of the plane ZOP relative to some fixed plane through OZ.

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  • R is the radial component of force, is the moment about a lilie through 0 perpendicular to the plane ZOP, and 4 is the moment about OZ.

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  • Again, a vertical plane passing, through O and a point where the motion is horizontal is evidently a plane of symmetry as regards the path.

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  • The meaning of these quantities is easily recognized; thus X is the angular momentum about a horizontal axis normal to the plane of 0, u is the angular momentum about the vertical OZ, and s is the angular momentum about the axis of symmetry..

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  • For simplicity we will suppose that the motion is confined to one vertical plane.

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  • Relations between Polygons of Loads and of Resistances.In a structure in which each piece is supported at two joints only, the well-known laws of statics show that the directions of the gross load on each piece and of the two resistances by which it is supported must lie in one plane, must either be parallel or meet in one point, and must bear to each other, if not parallel, the proportions of the sides of a triangle respectively parallel to their directions, and, if parallel, such proportions that each of the three forces shall be proportional to the distance between the other two,all the three distances being measured along one direction.

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  • Further, at any one of the centres of load let PL represent the magnitude and direction of the gross load, and Pa, Pb the two resistances by which the piece to which that load is applied is supported; then wifl those three lines be respectively the diagonal and sides of a parallelogram; or, what is the same thing, they will be equal to the three sides of a triangleS and they must be in the same plane, although the sides of the polygon of resistances may be in different planes.

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  • Let a represent the area of the section of a piston made by a plane perpendicular to its direction of motion, and v its velocity, which is to be considered as positive when outward, and negative when inward.

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  • Hence also the ratio of the com ponents of the velocities of two points A and B in the directions AP and BW respectively, both in the plane of rotation, is equal to the ratio of the perpendiculars Fni and Fn.

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  • The line T on the surface bbb has also for the instant no velocity in the plane AB; for it has just ceased to move towards the fixed surface aaa, and is just about to begin to move away from that surface.

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  • Let -y denote the total angular velocity of the rotation of the cone B about the instantaneous axis, $ its angular velocity about the axis OB relatively to the plane AOB, and a the angular velocity with which the plane AOB turns round the axis OA.

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  • Let yr be the linear velocity of the point E fixed in the plane of axes AOB.

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  • Now, as the line of contact OT is for the instant at rest on the rolling cone as well as on the fixed cone, the linear velocity of the point E fixed to the plane AOB relatively to the rolling cone is the same with its velocity relatively to the fixed cone.

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  • Then the motion of P is perpendicular to the plane OPQ, and its velocity is v,.= y.

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  • That, when both pieces rotate, their axes, and all their points of contact, lie in the same plane.

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  • That for a pair of turning pieces with parallel axes, and for a turning piece and a shifting piece, the line of contact is straight, and parallel to the axes or axis; and hence that the rolling surfaces are either plane or cylindrical (the term cylindrical including all surfaces generated by the motion of a straight line parallel tO itself).

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  • Cylindrical Wheels and Smooth Racks.In designing cylindrical wheels and smooth racks, and determining their comparati* motion, it is sufficient to consider a section of the pair of pieces made by a plane perpendicular to the axis or axes.

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  • The points where axes intersect the plane of section are called centres; the point where the line of contact intersects it, the poini of contact, or pitch-point; and the wheels are described as circular, elliptical, &c., according to the forms of their sections made by that plane.

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  • When the velocity ratio is variable, the line of contact will shift its position in the plane C1OC2, and the wheels will be cones, with eccentric or irregular bases.

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  • One of those consists in forming the rim of each wheel into a series of alternate ridges and grooves parallel to the plane of rotation; it is applicable to cylindrical and bevel wheels, but not to skew-bevel wheels.

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  • The pitch-circles of a pair of circular toothed wheels are sections of their pitch-surfaces, made for spur-wheels (that is, for wheels whose axes are parallel) by a plane at right angles to the axes, and for bevel wheels by a sphere described about the common apex.

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

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  • When the a,ces of a pair of pulleys are not parallel, the pulleys should be so placed that the part of the belt which is approaching each pulley shall be in the plane of the pulley.

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  • The velocity of the other connected point at such an instant is null, unless it also reaches a dead-point at the same instant, so that the line of connection is in the plane of the two axes of rotation, in which case the velocity ratio is indeterminate.

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  • If a connected point belongs to a turning piece, the direction of its motion at a given instant is perpendicular to the plane containing the axis and crank-arm of the piece.

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  • If a connected point belongs to a shifting piece, the direction of its motion at any instant is given, and a plane can be drawn perpendicular to that direction.

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  • To obviate this evil a short intermediate shaft is introduced, making equal angles with the first and last shaft, coupled with each of them by a Hookes joint, and having its own two forks in the same plane.

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  • These rolling cylinders are sometimes called axodes, and a section of an axode in a plane parallel to the plane of motion is called a centrode.

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  • The axode is hence the locus of the instantaneous axis, whilst the centrode is the locus of the instantaneous centre in any plane parallel to the plane of motion.

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  • To find the form of these surfaces corresponding to a particular pair of non-adjacent links, consider each link of the pair fixed in turn, then the locus of the instantaneous axis is the axode corresponding to the fixed link, or, considering a plane of motion only, the locus of the instantaneous centre is the ceotrode corresponding to the fixed link.

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  • The flat pivot is a cylinder of steel having a plane circular end as a rubbing surface.

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  • Should the pen have a nib of two jaws, like those of an ordinary drawing-pen, the plane of the jaws must pass through PT.

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  • Friction of Teeth.Let N be the normal pressure exerted between a pair of teeth of a pair of wheels; s the total distance through which they slide upon each other; n the number Of pairs of teeth which pass the plane of axis in a unit of time; then nf NI (63)

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  • Centrifugal Force of a Rotating BodyThe centrifugal force exerted by a rotating body on its axis of rotation is the same in magnitude as if the mass of the body were concentrated at its centre of gravity, and acts in a plane passing through the axis of rotation and the centre of gravity of the body.

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  • The plane through 0 to which the shaft is perpendicular is called the reference plane, because all the transferred forces act in that plane at the point 0.

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  • The plane through the radius of the weight containing the axis OX is railed the axial plane because it contains the forces forming the couple due to the transference of F to the reference plane.

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  • For the vector representing a couple of the type War, if the masses are all on the same side of the reference plane, the direction of drawing is from the axis outwards; if the masses are some on one side of the reference plane and some on the other side, the direction of drawing is from the axis outwards towards the weight for all masses on the one side, and from the mass inwards towards the axis for all weights on the other side, drawing always parallel to the direction defined by the radius r.

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  • To solve this the reference plane must be chosen so that ii coincides with the plane of revolution of one of the as yet unknowr balance weights.

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  • The balance weight in this plane has therefon no couple corresponding to it.

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  • Then, transferring the product Wr correspondinf with this balance weight to the reference plane, proceed to draw the force polygon.

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  • The vector required to close it will determine the second balance weight, the work may be checked by taking the reference plane to coincide with the plane of revolution of the second balance weight and then re-determining them, or by taking a reference plane anywhere and including the two balance weights trying if condition (c) is satisfied.

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  • Although this method balances the pistons in the horizontal plane, and thus allows the pull of the engine on the train to be exerted without the variation due to the reciprocation of the pistons, yet the force balanced horizontally is introduced vertically and appears as a variation of pressure on the rail.

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  • According to the principles of statics, the resultant of the force P, applied at G perpendicular to the plane OG, and the couple M is a force equal and parallel to P, but applied at a distance GC from G, in the prolongation of the perpendicular OG, whose value is GC = M/P = R2/OG.

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  • Then its motion may be analysed into (I) a translation of its centre of gravity; and (2) a rotation about an axis through its centre of gravity perpendicular to its plane of motion.

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  • Adopting the hypothesis of two fluids, Coulomb investigated experimentally and theoretically the distribution of electricity on the surface of bodies by means of his proof plane.

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  • He determined the law of distribution between two conducting bodies in contact; and measured with his proof plane the density of the electricity at different points of two spheres in contact, and enunciated an important law.

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  • The 19th series (1845) contains an account of his brilliant discovery of the rotation of the plane of polarized light by transparent dielectrics placed in a magnetic field, a relation which established for the first time a practical connexion between the phenomena of electricity and light.

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  • Two of these rollers are supported in the same horizontal plane of the framework, while the third or top roller is kept in close contact by means of weights and springs acting on each end of the arbor.

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  • It consisted of a plane conducting plate forming one pan of a balance which was suspended over another insulated plate which could be electrified.

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  • All that at present can be attempted is, to reproduce a single plane in another plane; but even this has not been altogether satisfactorily accomplished, aberrations always occur, and it is improbable that these will ever be entirely corrected.

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  • We receive, therefore, in no single intercepting plane behind the system, as, for example, a focussing screen, an image of the object point; on the other hand, in each of two planes lines 0' and 0" are separately formed (in neighbouring planes ellipses are formed), and in a plane between 0' and 0" a circle of least confusion.

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  • Two " astigmatic image surfaces " correspond to one object plane; and these are in contact at the axis point; on the one lie the focal lines of the first kind, on the other those of the second.

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  • The course of the rays in the meridional section is no longer symmetrical to the principal ray of the pencil; and on an intercepting plane there appears, instead of a luminous point, a patch of light, not symmetrical about a point, and often exhibiting a resemblance to a comet having its tail directed towards or away from the axis.

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  • If the above errors be eliminated, the two astigmatic surfaces united, and a sharp image obtained with a wide aperture - there remains the necessity to correct the curvature of the image surface, especially when the image is to be received upon a plane surface, e.g.

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  • If, in an unsharp image, a patch of light corresponds to an object point, the " centre of gravity " of the patch may be regarded as the image point, this being the point where the plane receiving the image, e.g.

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  • The origins of these four plane co-ordinate systems may be collinear with the axis of the optical system; and the corresponding axes may be parallel.

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  • On account of the aberrations of all rays which pass through 0, a patch of light, depending in size on the lowest powers of E, x, y which the aberrations contain, will be formed in the plane I'.

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  • The existence of an optical system, which reproduces absolutely a finite plane on another with pencils of finite aperture, is doubtful; but practical systems solve this problem with an accuracy which mostly suffices for the special purpose of each species of instrument.

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  • The problem of finding a system which reproduces a given object upon a given plane with given magnification (in so far as aberrations must be taken into account) could be dealt with by means of the approximation theory; in most cases, however, the analytical difficulties are too great.

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  • Spherical aberration and changes of the sine ratios are often represented graphically as functions of the aperture, in the same way as the deviations of two astigmatic image surfaces of the image plane of the axis point are represented as functions of the angles of the field of view.

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  • For example, it is possible, with one thick lens in air, to achromatize the position of a focal plane of the magnitude of the focal length.

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  • In a plane containing the image point of one colour, another colour produces a disk of confusion; this is similar to the confusion caused by two " zones " in spherical aberration.

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  • It was not too late to arrest the Galatians on their downward plane, and the apostle, unable or unwilling to re-visit them, despatched this epistle.

    0
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  • Laevo-tartaric acid is identical in its chemical and in most of its physical properties with the dextro-acid, differing chiefly in its action on polarized light, the plane of polarization being rotated to the left.

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  • The optic figure seen in convergent polarized light through a section cut parallel to the plane of symmetry of a borax crystal is symmetrical only with respect to the central point.

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  • Milner's own object in assenting to the introduction of the Chinese was - besides aiding to put the gold mining industry on a more stable basis - to obtain revenue for the great task he had on hand, " the restarting of the colonies on a higher plane of civilization than they had ever previously attained "; and in respect of the working of the mines and consequently in providing revenue the introduction of the Chinese proved eminently successful; but in February 1906 the Campbell-Bannerman administration felt it incumbent to announce that no ordinance imposing " servile conditions " would be sanctioned.

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  • Distinct crystals are rarely met with; these are rhombohedral and isomorphous with arsenic and bismuth; they have a perfect cleavage parallel to the basal plane, c (111), and are sometimes twinned on a rhombohedral plane, e (1 ro).

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  • In Kashmir the plane and Lombardy poplar flourish, though hardly seen farther east, the cherry is cultivated in orchards, and the vegetation presents an eminently European cast.

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  • At the entrance of the town stands a noble chinar (oriental plane), measuring 45 ft.

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  • Their conventionality sets the lyrics of Cruz e Silva on a lower plane, but in the Hyssope he improves on the Lutrin of Boileau.

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  • In attempting to calculate the effect of this surface-tension in determining the form of a drop of the liquid, Segner took account of the curvature of a meridian section of the drop, but neglected the effect of the curvature in a plane at right angles to this section.

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  • Let the plane of the paper be a normal section of the surface of the stratum at the point B, making an angle w with the FIG.

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  • If x' is the potential energy of unit of mass of the substance in vapour, then at a distance z from the plane surface of the liquid X = X' - 22 7rp 7rpe e ((zo)) ..

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  • Now 2 7rmpi,t(c) represents the attraction between a particle m and the plane surface of an infinite mass of the liquid, when the distance of the particle outside the surface is c. Now, the force between the particle and the liquid is certainly, on the whole, attractive; but if between any two small values of c it should be repulsive, then for films whose thickness lies between these values the tension will increase as the thickness diminishes, but for all other cases the tension will diminish as the thickness diminishes.

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  • The former can be found at once by calculating the mutual attraction of the parts of a large mass which lie on opposite sides of an imaginary plane interface.

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  • If the density be a, the attraction between the whole of one side and a layer upon the other distant z from the plane and of thickness dz is 27r6 2 P(z)dz, reckoned per unit of area.

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  • If we now suppose the crevasse produced by direct separation of its walls, the work necessary must be the same as before, the initial and final configurations being identical; and we recognize that the tension may be measured by half the work that must be done per unit of area against the mutual attraction in order to separate the two portions which lie upon opposite sides of an ideal plane to a distance from one another which is outside the range of the forces.

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  • If four fluids, a, b, c, d, meet in a point 0, and if a tetrahedron AB CD is formed so that its edge AB represents the tension of the surface of contact of the liquids a and b, BC that of b and c, and so on; then if we place this tetrahedron so that the face ABC is normal to the tangent at 0 to the line of concourse of the fluids abc, and turn it so that the edge AB is normal to the tangent plane at 0 to the surface of contact of the fluids a and b, then the other three faces of the tetrahedron will be normal to the tangents at 0 to the other three lines of concourse of the liquids, an the other five edges of the tetrahedron will be normal to the tangent planes at 0 to the other five surfaces of contact.

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  • Hence if we take two nets of wire with hexagonal meshes, and place one on the other so that the point of concourse of three hexagons of one net coincides with the middle of a hexagon of the other, and if we then, after dipping them in Plateau's liquid, place them horizontally, and gently raise the upper one, we shall develop a system of plane laminae arranged as the walls and floors of the cells are arranged in a honeycomb.

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  • The work required to cleave asunder the parts of the first fluid which lie on the two sides of an ideal plane passing through the interior, is per unit of area 2T 1, and the free surface produced is two units in area.

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  • Let l be the breadth of the plates measured perpendicularly to the plane of the paper, then the length of the line which bounds the wet and the dry parts of the plates inside is 1 for each surface, and on this the tension T acts at an angle a to the vertical.

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  • If we suppose these generating lines to be normal to the plane of the paper, then all sections of the solids parallel to this plane will be equal and similar to each other, and the section of the surface of the liquid will be of the same form for all such sections.

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  • In this case drops which break away with different velocities are carried under the action of gravity into different paths; and thus under ordinary circumstances a jet is apparently resolved into a " sheaf," or bundle of jets all lying in one vertical plane.

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  • We know that the radius of curvature of a surface of revolution in the plane normal to the meridian plane is the portion of the normal intercepted by the axis of revolution.

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  • The simplest case is that of a rectangular orifice in a horizontal plane, the sides being a and b.

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  • Let the surface of separation be originally in the plane of the orifice, and let the co-ordinates x and y be measured from one corner parallel to the sides a and b respectively, and let z be measured upwards.

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  • In order to render visible the small waves employed, and which we may regard as deviations of a plane surface from its true figure, the method by which Foucault tested reflectors is suitable.

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  • His method was that of observing the form of a large drop standing on a plane surface.

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  • If K is the height of the flat surface of the drop, and k that of the point where its tangent plane is vertical, then T = 1(K - k) 2gp. Quincke finds that for several series of substances the surfacetension is nearly proportional to the density, so that if we call Surface-Tensions of Liquids at their Point of Solidification.

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  • Again he fully accepts the influence of the stars on the production of the metals, whereas the Latin Geber disputes it, and in general the chemical knowledge of the two is on a different plane.

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  • The ship, in virtue of its being immersed in two fluids having different densities, can be steered and made to tack about in a horizontal plane in any given direction.

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  • The balloon in the absence of wind can only rise and fall in a vertical line; the flying creature can fly in a horizontal plane in any given direction.

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  • In this process the weight of the body performs an important part, by acting upon the inclined planes formed by the wings in the plane of progression.

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  • Borelli was of opinion that flight resulted from the application of an inclined plane, which beats the air, and which has a wedge b action.

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  • In the 1 96th proposition of his work (De motu animalium, Leiden, 1685) he states that " If the expanded wings of a bird suspended in the air shall strike the undisturbed air beneath it with a motion perpendicular to the horizon, the bird will fly with a transverse motion in a plane parallel with the horizon."

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  • He is of opinion that the insect abstracts from the air by means of the inclined plane a component force (composant) which it employs to support and direct itself.

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  • It will of its own accord dispose itself as an inclined plane, and receiving obliquely the reaction of the air, it transfers into tractile force a part of the vertical impulsion it has received.

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  • The machine, fully prepared for flight, was started from the top of an inclined plane, in descending which it attained a velocity necessary to sustain it in its further progress.

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  • The aeroplanes are aeronaut attached to main kept in parallel plane by spar.

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  • These investigators began their work in 1900, and at an early stage introduced two characteristic features - a horizontal rudder in front for steering in the vertical plane, and the flexing or bending of the ends of the main supporting aeroplanes as a means of maintaining the structure in proper balance.

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  • At last he came (here some words are missing) and began to teach sitting under a plane tree.

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  • The plane of the ecliptic is that plane in or near which the centre of gravity of the earth and moon.

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  • The ecliptic itself is the great circle in which this plane meets the celestial sphere.

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  • Owing to the action of the moon on the earth, as it performs its monthly revolution in an orbit slightly inclined to the ecliptic, the centre of the earth itself deviates from the plane of the ecliptic in a period equal to that of the nodal revolution of the moon.

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  • Owing to the action of the planets, especially Venus and Jupiter, on the earth, the centre of gravity of the earth and moon deviates by a yet minuter amount, generally one or two tenths of a second, from the plane of the ecliptic proper.

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  • The obliquity of the ecliptic is the angle which its plane makes with that of the equator.

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  • The ramparts are strengthened by two massive towers containing an inclined plane on which horses and carriages may ascend.

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  • Antlers arising at acute angles to the median line of the skull (as in the following genera), at first projecting from the plane of the forehead, and then continued upwards nearly in that plane, supported on short pedicles, and furnished with a brow-tine, never regularly forked at first division, but generally of large size, and with not less than three tines; the skull without ridges on the frontals forming the bases of the pedicles of the antlers.

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  • Wapiti, on the other hand, show a marked tendency to the flattening of the antlers, with a great development of the fourth tine, which is larger than all the others, and the whole of the tines above this in the same plane, or nearly so, this plane being the same as the long axis of the animal.

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  • The hangul (C. cashmirianus) of Kashmir is a distinct dark-coloured species, in which the antlers tend to turn in at the summit; while C. yarcandensis, of the Tarim Valley, Turkestan, is a redder animal, with a wholly rufous tail, and antlers usually terminating in a simple fork placed in a transverse plane.

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  • Lastly C. albirostris, of Tibet, is easily recognized by its white muzzle, and smooth, whitish, flattened antlers, which have fewer tines than those of the other members of the group, all placed in one plane.

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  • The pistil consists of a single carpel, opposite the pale in the median plane of the spikelet.

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  • The basal plane, so common on calcite and many other rhombohedral minerals, is of the greatest rarity in quartz, and when present only appears as a small rough face formed by the corrosion of the crystal.

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  • Usually they are interpenetration twins with the principal axis as twin-axis; the prism planes of the two individuals coincide, and the faces r and z also fall into the same plane.

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  • A ray of plane-polarized light traversing a right-handed crystal of quartz in the direction of the triad axis has its plane of polarization rotated to the right, while a left-handed crystal rotates it to the left.

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  • It consists of a single inclined plane stretching upwards, with a north-westerly direction, from the left bank of the river to the summits of the Carpathians.

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  • Such examples show the importance of placing any rain-gauge, so far as possible, upon a plane surface of the earth - horizontal, or so inclined that, if produced, especially in the direction of prevailing winds, it will cut the mean levels of the area whose mean rainfall is intended to be represented by that gauge.

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  • In this case, as in that of a level plane of uniformly porous sand, the vertical section of the figure is tangential to the vertical well and to the natural level of the subsoil water.

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  • The clay roof, rather than the walls of this crevice of sand, gave way and pressed down to fill the vacancy, and the leakage worked up along the weakened plane of tangential strain bd.

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  • Unless such places are carefully dug out or re-puddled before the work of filling is resumed, the percolation may increase along the vertical plane where it is greatest, by the erosion and falling in of the clay roof, as in the other cases cited.

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  • It may do this in virtue of horizontal water-pressure alone, or of such pressure combined with upward pressure from intrusive water at its base or in any higher horizontal plane.

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  • One such assumption is that, if the dam is well built, the intensity of vertical pressure will (neglecting local irregularities) vary nearly uniformly from face to face along any horizontal plane.

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  • Hence it follows that on the assumption of uniformly varying stress the line of pressures, when the reservoir is full, should not at any horizontal plane fall outside the middle third of the width of that plane.

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  • Then the difference between the normal pressure on a rectangular element in the lower plane and that on the upper plane is the weight of the element and the difference between the shears on the vertical faces of that element.

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  • The late Sir Benjamin Baker, F.R.S., suggested that the stresses might be measured by experiments with elastic models, and among others, experiments were carried out by Messrs Wilson and Gore a with indiarubber models of plane sections of dams (including the foundations) who applied forces to represent the gravity and water pressures in such a manner that the virtual density of the rubber was increased many times without interfering with the proper ratio between gravity and water pressure, and by this means the strains produced were of sufficient magnitude to be easily measured.

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  • As a result of this theory, in the case of a retaining wall supporting a vertical face of earth beneath an extended horizontal plane level with the top of the wall, we get p _ wx 2 1 - sin ii 2 I +sin P' [[Reservoir Empty Reservoir Full Ellipses Of Vertical Pressures On Horizontal Joints]].

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  • The materials, however, were poor, and it is probable that rupture by tension in a roughly horizontal plane took place.

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  • It is clear that errors will arise if the pieces of steel are not truly perpendicular to the plane of the beam, and the adjust - ment of great accuracy would be very tedious.

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  • For this purpose a number of separate weighbridges of simple construction are erected, one for each wheel of the engine, with their running surfaces in exactly the same horizontal plane.

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  • E, Section through a dividing calicle of Mussa, showing the union of two septa in the plane of division, and the origin of new septa at right angles to them.

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  • This can be accomplished by attaching balance-weights to the pulley until it will remain stationary in all positions, when its shaft rests on two horizontal knife-edges in the same horizontal plane, or, preferably, the pulley and shaft may be supported on bearings resting on springs, and balanced by attached masses until there is no perceptible vibration of the springs at the highest speed of rotation.

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  • The rims of pulleys, round which flat bands are wrapped, may be truly cylindrical, in which case the belt will run indifferently at any part of the pulley, or the rim may be swelled towards the centre, when the central line of the band will tend to run in the diametral plane of the pulley.

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  • When pulleys are mounted on shafts which are parallel to one another, the band will retain its position, provided that its central line advances towards each pulley in the diametral plane of this latter.

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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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  • In the majority of folds the bending of the strata has taken place about an "axial plane" (often called the "axis"), which in the examples illustrated in fig.

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  • One of the important functions of a fold is its direction; this of course depends upon the orientation of the axial plane.

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  • Moreover, unlike his Danish predecessor, he looked down upon the English from the plane of a higher civiliza- tion; the Normans regarded the conquered nation as barbarous and boorish.

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  • Here is the great cataract of Itamaraca, which rushes down an inclined plane for 3 m.

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

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  • The foregoing notion of a point at infinity is a very important one in modern geometry; and we have also to consider the paradoxical statement that in plane geometry, or say as regards the plane, infinity is a right line.

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  • If with a given centre of projection, by drawing from it lines to every point of a given line, we project the given line on a given plane, the projection is a line, i.e.

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  • Say the projection is always a line, then if the figure is such that the two planes are parallel, the projection is the intersection of the given plane by a parallel plane, or it is the system of points at infinity on the given plane, that is, these points at infinity are regarded as situate on a given line, the line infinity of the given plane.

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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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  • In this memoir by Gergonne, the theory of duality is very clearly and explicitly stated; for instance, we find " da p s la geometrie plane, a chaque theoreme ii en repond necessairement un autre qui s'en deduit en echangeant simplement entre eux les deux mots points et droites; tandis que dans la geometrie de l'espace ce sont les mots points et plans qu'il faut echanger entre eux pour passer d'un theoreme a son correlatif "; and the plan is introduced of printing correlative theorems, opposite to each other, in two columns.

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

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

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  • Stated in regard to the cone, we have there the fundamental theorem that there are two different kinds of sheets; viz., the single sheet, not separated into two parts by the vertex (an instance is afforded by the plane considered as a cone of the first order generated by the motion of a line about a point), and the double or twin-pair sheet, separated into two parts by the vertex (as in the cone of the second order).

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  • It may be mentioned that the single sheet is a sort of wavy form, having upon it three lines of inflection, and which is met by any plane through the vertex in one or in three lines; the twin-pair sheet has no lines of inflection, and resembles in its form a cone on an oval base.

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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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  • An even circuit not cutting itself divides the plane into two parts, the one called the internal part, incapable of containing any odd circuit, the other called the external part, capable of containing an odd circuit.

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

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