Plane sentence example

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

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

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

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

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  • Hence in any infinitesimal part of the fluid the circulation is zero round every small plane curve passing through the vortex line; and consequently the circulation round any curve drawn on the surface of a vortex filament is zero.

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

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  • For absolute determinations of wave-lengths plane gratings are used.

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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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  • 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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  • 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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  • At the southern end of the boulevard de la Republique is the square de la Republique, formerly the place Bresson, in which is the municipal theatre; at the other extremity of the boulevard is the place du Gouvernement, which is planted on three sides with a double row of plane trees and is the fashionable resort for evening promenade.

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  • Transporters can only move the load to any point on a vertical surface (generally a plane surface); they have a lifting Trans- motion and a movement of translation.

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  • The indicator was connected with a Ruhmkorff coil or other equivalent apparatus, designed to cause a continual succession of sparks to pass between the indicator and a metal plate situated beneath it and having a plane surface parallel to its line of motion.

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  • The short leg of the siphon tube dips into an insulated ink-bottle, so' that the ink it contains becomes electrified, while the long leg has its open end at a very small distance from a brass table, placed with its surface parallel to the plane in which the mouth of the leg moves, and over which a slip of paper may be passed at a uniform rate, as in the spark recorder.

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  • In this way are formed the familiar feathery colonies of Plumularia, in which the pinnules are all in one plane, while in the allied Antennularia the pinnules are arranged in whorls round the main biserial stem.

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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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  • These may be looked The six upon as being all derived by various modifications or elementa ry arrangements of the single form-unit, the slope or inclined land forms. plane surface.

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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 plan