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rotation

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rotation

rotation Sentence Examples

  • The rotation of the earth thus fixes the directions of north and south and defines those of east and west.

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  • Such rotation can obviously be controlled within limits that need not be further considered.

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  • The opportunities which rotation cropping affords for the cleaning of the land from weeds is another distinct element of advantage.

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  • Under hay are included the produce of clover, sainfoin and rotation grasses, and also that of permanent meadow.

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  • The cups are placed symmetrically on the end of the arms, and it is easy to see that the wind always has the hollow of one cup presented to it; the back of the cup on the opposite end of the cross also faces the wind, but the pressure on it is naturally less, and hence a continual rotation is produced; each cup in turn as it comes round providing the necessary force.

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  • The hay made from clover, sainfoin and grasses under rotation generally gives a bigger average yield than that from permanent grass land.

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  • Already anxieties appear as to the theological verdict upon two of his fundamental views - the infinitude of the universe, and the earth's rotation round the sun.

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  • Thus the smallest rotation of either head communicates to the corresponding slide motion, which, if the screws are accurate, is proportional to the amount through which the head is turned.

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  • The rotation of the planets on their axes is also explained as a consequence of the nebular theory, for at the time of the first formation of the planet it must have participated in the rotation of the whole nebula, and by the subsequent contraction of the planet the speed with which the rotation was performed must have been accelerated.

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  • But the average produce over forty years of continuous growth of barley was, in all cases where nitrogenous and mineral manures (containing phosphates) were used together, much higher than the average produce of the crop grown in ordinary rotation in the United Kingdom, and very much higher than the average in most other countries when so grown.

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  • Remarkable as Hellriegel's discovery was, it merely furnished the explanation of a fact which had been empirically established by the husbandman long before, and had received most intelligent application when the old four-course (or Norfolk) rotation was devised.

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  • In whatever form energy is produced and distributed to the train it ultimately appears as mechanical energy applied to turn one or more axles against the resistance to their rotation imposed by the weight on the wheels and the motion of the train.

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  • The metropolitan of Athens is president, and there are four other members appointed by the government in annual rotation from the senior bishops.

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  • Whatever the specific rotation, there may in practice be deviations from the plan of retaining on the farm the whole of the root-crops, the straw of the grain crops and the leguminous fodder crops (clover, vetches, sainfoin, &c.) for the production of meat or milk, and, coincidently, for that of manure to be returned to the land.

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  • This gigantic nebulous mass, of which the sun was only the central and somewhat more condensed portion, is supposed to have a movement of rotation on its axis.

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  • A similar rotation and dislocation occurs in various petrels, in correlation with the indigestible sepia-bills, &c., which these birds swallow in great quantities.

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  • Of lime, very little is taken up by the cereal crops, and by the root-crops much less than of potash; more by the leguminous than by the other crops, and, by the clover especially, sometimes much more than by all the other crops of the rotation put together.

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  • In the case of plants the method of procedure was to grow some of the most important crops of rotation, each separately year after year, for many years in succession on the same land, (a) without manure, (b) with farmyard manure and (c) with a great variety of chemical manures; the same description of manure being, as a rule, applied year after year on the same plot.

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  • Experiments on an actual course of rotation, without manure, and with different manures, have also been made.

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  • Whilst much grass land has been laid down with the intention from the outset that it should be permanent, at the same time some considerable areas have through stress of circumstances been allowed to drift from the temporary or rotation grass area to the permanent list, and have thus still further diminished the area formerly under the dominion of the plough.

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  • Another important consequence of rotation is the deviation produced in moving bodies relatively to the surface.

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  • The benefits that accrue from the practice of rotation are well illustrated in the results obtained from the investigations at Rothamsted into the simple four-course system, which may fairly be regarded as a self-supporting system.

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  • The greater freedom of cropping and the less close adherence to the formal system of rotation of crops, which characterize the early years of the 10th century, rest upon a scientific basis.

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  • Cereal pests can only be treated by general cleanliness and good farming, and of course they are largely kept down by the rotation of crops.

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  • The diurnal rotation of the earth furnishes two fixed points or poles, the axis joining which is fixed or nearly so in its direction in space.

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  • Although many different rotations of crops are practised, they may for the most part be considered as little more than local adaptations of the system of alternating root-crops and leguminous crops with cereal crops, as exemplified in the old four-course rotation - roots, barley, clover, wheat.

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  • Of phosphoric acid, the cereal crops take up as much as, or more than, any other crops of the rotation, excepting clover; and the greater portion thus taken up is lost to the farm in the saleable product - the grain.

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  • In the form known as Ferrell's Law this runs: " If a body moves in any direction on the earth's surface, there is a deflecting force which arises from the earth's rotation which tends to deflect it to the right in the northern hemisphere but to the left in the southern hemisphere."

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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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  • This cone is driven by gearing from the wire drum, so that it rotates at the speed of the outgoing wire, the direction of rotation being such as to cause the nut to travel towards the smaller end of the cone.

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  • Repetition of one crop in the field exhausts the ground; but rotation of the crops is good for the soil.

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  • In fact a little consideration of the theory of probabilities will show it to be infinitely probable that such an object should really have some movement of rotation, no matter by what causes the nebula may have originated.

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  • Of potash, each of the rotation crops takes up very much more than of phosphoric acid.

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  • wheat 26�08 bushels, barley 2 9.3 o bushels, oats 38.14 bushels, beans 19.61 bushels, rotation hay 23.55 cwt., permanent hay 20.41 cwt.

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  • But much less potash than phosphoric acid is exported in the cereal grains, much more being retained in the straw, whilst the other products of the rotation - the root and leguminous crops - which are also supposed to be retained on the farm, contain very much more potash than the cereals, and comparatively little of it is exported in meat and milk.

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  • Thus the cereal crops, when grown in rotation, yield more produce for sale in the season of growth than when grown continuously.

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  • In the rotation of crops there is an amazing diversityshifts of two years, three years, four years, six years, and in many cases whatever order strikes the fancy of the farmer.

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  • Under the old Norfolk or four-course rotation (roots, barley, clover, wheat) land thus seeded with clover or grass seeds was intended to be ploughed up at the end of a year.

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  • Very little of the lime of the crops, however, goes off in the saleable products of the farm in the case of the self-supporting rotation under consideration.

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  • Thus the whole of the crops of rotation take up very much more of potash than of phosphoric acid, whilst probably even less of it is ultimately lost to the land.

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  • Sent her to half a dozen agencies on rotation.

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  • The landlord lets his land to two or more persons jointly, who undertake to restore it to him in good condition with one-third of it interrozzito, that is, fallow, so as to be cultivated the following year according to triennial rotation.

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  • Protoplasmic Movements.In the cells of many plants the cytoplasm frequently exhibits movements of circulation or rotation.

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  • speed to 60 fathoms for 20 knots, the pull of the line and rotator is borne by coned rollers, having their outlines tapering to a common point in their rotation, thus giving a broad rolling surface.

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  • There is no difficulty in conceiving how a nebula, quite independently of any internal motion of its parts, shall also have had as a whole a movement of rotation.

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  • of hay per acre less than that from clover, sainfoin and grasses under rotation.

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  • In one case, indeed, the average produce by mixed minerals and nitrogenous manure was more than that by the annual application of farmyard manure; and in seven out of the ten cases in which such mixtures were used the average yield per acre was from over two to over eight bushels more than the average yield of the United Kingdom (assuming this to be about twenty-eight bushels of 60 lb per bushel) under ordinary rotation.

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  • It is not only the conditions of growth, but the uses to which the different crops are put, that have to be considered in the case of rotation.

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  • They are effected chiefly by some alteration in the description of the root-crop, and perhaps by the introduction of the potato crop; by growing a different cereal, or it may be more than one cereal consecutively; by the growth of some other leguminous crop than clover, since " clover-sickness " may result if that crop is grown at too short intervals, or the intermixture of grass seeds with the clover, and perhaps by the extension by one or more years of the period allotted to this member of the rotation.

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  • Reference may first be made to the important mineral constituents of different crops of the four-course rotation.

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  • - Sketch of a model designed so as to show the effect of torsion or rotation of the visceral hump in Streptoneurous Gastropoda.

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  • Neither the rotation of the shell as a whole nor its helicoid spiral coiling is the immediate cause of the torsion of the body in the individual, for the direction of the torsion is indicated in the segmentation of the ovum, in which there is a complete A B From Lankester's Treatise on Zoology.

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  • Nothing definite can be said with regard to a rotation of crops Sea Island Cotton - Carolina Sea Island Florida „ „ Georgia „ Barbados „ „ Egyptian Cottons Yannovitch.

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  • It is believed that the rotation must differ with every variety of soil, with the result that each planter has his own method, and little can be said in general.

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  • A more careful study of the physical as well as the chemical properties of a soil must precede intelligent experimentation in rotation.

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  • Fields considered utterly used up, and allowed to " rest " for years, when cultivated again have produced better crops than those which had been under a more or less thoughtful rotation.

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  • The following rotation is frequently adopted.

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  • This may consist simply in attaching one end of the wire to an index lever and the other to a fixed support, or the elongation of the wire may cause a rotation in a mirror from which a ray of light is reflected, and the movement of this ray over a scale will then provide the necessary means of indication.

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  • The method is based on the supposition that the magnetic rotation measures the strain produced in the molecule by an auxochrome, and he arranges the groups in the following order: O.

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  • This main scheme is complicated in various ways: (r) by the rotation of the earth, which continually deflects currents of water or air to the right in the northern or to the left in the southern hemisphere; (2) by the conformation of the land masses (as in the case of the equatorial stream which is banked up in the Gulf of Mexico and flows out through the Straits of Florida); (3) by the varying depth of the ocean, for currents tend to flow more readily through deep than in shallow waters (as in the case of the main Atlantic drift, which flows most strongly through the deep channel between Shetland and the Faroe Is.); and (4) by the driving force of the winds acting on the surface of the sea (thus the drift of water from the equator is not N.E., as one might expect, but from E.

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  • It differs from egg-albumin in its specific rotation (-57° to - 64°), and in being slowly coagulated by alcohol and ether.

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  • It coagulates at about 56°, and its specific rotation is - 30.70°.

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  • They are dextrorotatory, and the specific rotation is numerically greater than that of albumin; hence the proteids are, in general, dextrorotatory.

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  • When heated to above 200 it turns brown and produces caramel, a substance possessing a bitter taste, and used, in its aqueous solution or otherwise, under various trade names, for colouring confectionery, spirits, &c. The specific rotation of the plane of polarized light by glucose solutions is characteristic. The specific rotation of a freshly prepared solution is 105°, but this value gradually diminishes to 52.5°, 24 hours sufficing for the transition in the cold, and a few minutes when the solution is boiled.

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  • The specific rotation also varies with the concentration; this is due to the dissociation of complex molecules into simpler ones, a view confirmed by cryoscopic measurements.

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  • by determining the rotation of the plane of polarization of a solution, or, chemically, by taking advantage of its property of reducing alkaline copper solutions.

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  • Hence processes have been patented in which the objects to be plated are suspended in revolving drums between the anodes, the rotation of the drum causing the constant renewal of surfaces and affording a burnishing action at the same time.

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  • This is called the direct orthogonal substitution, because the sense of rotation from the axis of X i to the axis of X, is the same as that from that of x i to that of x 2.

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  • If the senses of rotation be opposite we have the skew orthogonal substitution x1 =cos0Xi+sinOX2r x 2 = sin °Xicos OX2r of modulus -1.

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  • the effect of parallax could be obtained as well as by observing from two different stations; in fact the rotation of the earth carried the observer himself round a parallel of latitude, so that the comparison of his own morning and evening observations could be used as if they had been made at different stations.

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  • 3 The lines of force are evidently circles concentric with the wire and at right angles to it; their direction is related to that of the current in the same manner as the rotation of a corkscrew is related to its thrust.

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  • The field at the centre of a circular conductor of radius r through which current is passing is H = 27ri/r, (3) the direction of the force being along the axis and related to the direction of the current as the thrust of a corkscrew to its rotation.

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  • The direction of the force is parallel to the axis of the coil, and related to the direction of the current as the thrust of a corkscrew to its rotation.

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  • When the specimen rotates, the magnet is deflected from its upright position by an amount which depends upon the work done in a single complete rotation, and therefore upon the hysteresis.

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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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  • If the field is uniform, H=O/wd, where 0 is the rotation, d the thickness of the substance arranged as a plate at right angles to the direction of the field, and w Verdet's constant for the substance.

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  • As the source of monochromatic light a bright sodium burner is used, and the rotation, which is exactly proportional to H, is measured by an accurate polarimeter.

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  • A part of one surface of the plate may be silvered, so that the polarized ray, after having once traversed the glass, is reflected back again; the rotation is thus doubled, and moreover, the arrangement is, for certain experiments, more convenient than the other.

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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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  • There are strong reasons for believing that magnetism is a phenomenon involving rotation, and as early as 1876 Rowland, carrying out an experiment which had been proposed by Maxwell, showed that a revolving electric charge produced the same magnetic effects as a current.

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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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  • He was the author of four scientific treatises: Lezioni di fisica (2 vols., Pisa, 1841), Lezioni sui fenomeni fisicochimici dei corgi viventi (Pisa, 1844), Manuale di telegrafia elettrica (Pisa, 1850) and Cours special sur l'induction, le magnetisme de rotation, &c. (Paris, 1854).

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  • They relate almost entirely to electrical phenomena, such as the magnetic rotation of light, the action of gas batteries, the effects of torsion on magnetism, the polarization of electrodes, &c., sufficiently complete accounts of which are given in Wiedemann's Galvanismus.

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  • His treatise was remarkable, not only as offering a satisfactory explanation of the coincidence between the lunar periods of rotation and revolution, but as containing the first employment of his radical formula of mechanics, obtained by combining with the principle of d'Alembert that of virtual velocities.

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  • C. Oersted (1777-1851) had shown that a magnetic needle is deflected by an electric current, he attempted, in the laboratory of the Royal Institution in the presence of Humphry Davy, to convert that deflection into a continuous rotation, and also to obtain the reciprocal effect of a current rotating round a magnet.

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  • 4to, Paris, 1799) contains methods for calculating the movements of translation and rotation of the heavenly bodies, for determining their figures, and resolving tidal problems; the second, especially dedicated to the improvement of tables, exhibits in the third and fourth volumes (1802 and 1805) the application of these formulae; while a fifth volume, published in three instalments, 1823-1825, comprises the results of Laplace's latest researches, together with a valuable history of progress in each separate branch of his subject.

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  • It may be added that he first examined the conditions of stability of the system formed by Saturn's rings, pointed out the necessity for their rotation, and fixed for it a period (Io h 33 m) virtually identical with that established by the observations of Herschel; that he detected the existence in the solar system of an invariable plane such that the sum of the products of the planetary masses by the projections upon it of the areas described by their radii vectores in a given time is a maximum; and made notable advances in the theory of astronomical refraction (Mec. cel.

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  • A rotation by which one edge of the mirror advances 4X (while the other edge retreats to a like amount) introduces a phase-discrepancy of a whole period where before the rotation there was complete agreement.

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  • A rotation of this amount should therefore be easily visible, but the limits of resolving power are being approached; and the conclusion is independent of the focal length of the mirror, and of the employment of a telescope, provided of course that the reflected image is seen in focus, and that the full width of the mirror is utilized.

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  • A comparison with the method of a material pointer, attached to the parts whose rotation is under observation, and viewed through a microscope, is of interest.

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  • II), and it may often be compensated by a slight rotation in azimuth of the object-glass of the observing telescope.

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  • The error thus arising may be compensated by a rotation of the object-glass about one of the diameters y= =x.

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  • dr r In like manner we find TZ x d e ikr 2 - 471b 2 r dr From (to), (13), (24) we see that, as might have been expected, the rotation at any point is about an axis perpendicular both to the direction of the force and to the line joining the point to the source of disturbance.

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  • If the resultant rotation be n, we have TZ iJ (x 2 -{-y 2) de ikr TZsin4 d e ikr 2 r ' dr (r !

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  • TZsin r which completely determines the rotation at any point.

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  • The police courts of the City are held at the Mansion House, the Lord Mayor or an alderman sitting as magistrate, and at the Guildhall, where the aldermen preside in rotation.

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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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  • Putting (12) a vortex line is defined to be such that the tangent is in the direction of w, the resultant of, n, called the components of molecular rotation.

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  • If the liquid is stirred up by the rotation R of a cylindrical body, d4lds = normal velocity reversed dy = - Rx- Ry ds (5) ds 4' + 2 R (x2 + y2) = Y, (6) a constant over the boundary; and 4,' is the current-function of the relative motion past the cylinder, but now V 2 4,'+2R =o, (7) throughout the liquid.

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  • Example 3.-Analysing in this way the rotation of a rectangle filled with liquid into the two components of shear, the stream function 1//1 is to be made to satisfy the conditions v 2 /1 =0, 111+IRx 2 = IRa 2, or /11 =o when x= = a, +b1+IRx 2 = I Ra2, y ' 1 = IR(a 2 -x 2), when y = b Expanded in a Fourier series, 2 232 2 cos(2n+ I)Z?rx/a a -x 7r3 a Lim (2n+I) 3 ' (1) so that '?"

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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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  • Irrotational Motion in General.-Liquid originally at rest in a singly-connected space cannot be set in motion by a field of force due to a single-valued potential function; any motion set up in the liquid must be due to a movement of the boundary, and the motion will be irrotational; for any small spherical element of the liquid may be considered a smooth solid sphere for a moment, and the normal pressure of the surrounding liquid cannot impart to it any rotation.

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  • For a cavity filled with liquid in the interior of the body, since the liquid inside moves bodily for a motion of translation only, 41 = - x, 42 = -, 43 = - z; (2) but a rotation will stir up the liquid in the cavity, so that the'x's depend on the shape of the surface.

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  • When the liquid is bounded externally by the fixed ellipsoid A = A I, a slight extension will give the velocity function 4 of the liquid in the interspace as the ellipsoid A=o is passing with velocity U through the confocal position; 4 must now take the formx(1'+N), and will satisfy the conditions in the shape CM abcdX ¢ = Ux - Ux a b x 2+X)P Bo+CoB I - C 1 (A 1 abcdX, I a1b1cl - J o (a2+ A)P and any'confocal ellipsoid defined by A, internal or external to A=A 1, may be supposed to swim with the liquid for an instant, without distortion or rotation, with velocity along Ox BA+CA-B 1 -C1 W'.

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  • The extension of Green's solution to a rotation of the ellipsoid was made by A.

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  • As before in § 31, the rotation may be resolved into a shear-pair, in planes perpendicular to Ox and Oy.

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  • proceeds in a straight line, and the axis of rotation through the C.G.

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  • A rotation about the axis of a figure of revolution does not set the medium in motion, so that C 1 is.

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  • which is the ratio of the linear velocity of rotation 2dp to u, the velocity of advance, -T2 d2 C 22 tans = n 2 = 4 = (c 2 - Ct) cg C12 2 W!

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  • The British sovereign has 36 "Chaplains in Ordinary," who perform service at St James's in rotation, as well as "Honorary Chaplains" and "Chaplains of the Household."

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  • The rotation serves for the estimation of sugar solutions (saccharimetry).

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  • Also Kiliani found that the lactone derived from the cyanhydrin of natural arabinose (laevo) was identical with the previous lactone except that its rotation was equal and opposite.

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  • Its aqueous solution has a faint sweet taste, and is dextro-rotatory, the rotation of a fresh solution being about twice that of an old one.

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  • The estates are usually very large, and are divided up into fields which are cultivated in rotation, each field being given several years' rest after producing one crop. The tobacco is air-cured, fires being only employed during continuous wet weather, and the process of curing occupies four or five weeks.

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  • either (I) by the Mexican crusher or arrastra, in which the grinding is effected upon a bed of stone, over which heavy blocks of stone attached to cross arms are dragged by the rotation of the arms about a central spindle, or (2) by the Chilean mill or trapiche, also known as the edge-runner, where the grinding stones roll upon the floor, at the same time turning about a central upright - contrivances which are mainly used for the preparation of silver ores; but by far the largest proportion of the gold quartz of California, Australia and Africa is reduced by (3) the stamp mill, which is similar in principle to that used in Europe for the preparation of tin and other ores.

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  • Vegetables of all sorts are easily grown, and a rotation of these is raised on land irrigated from wells and springs.

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  • Some cotton is raised as a rotation crop, but no care is taken to improve the quality.

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  • When the rays of the sun or a candle, or dark radiation from a warm body, are incident on the vanes, the dark side of each vane is repelled more than the bright side, and thus the vanes are set into rotation with accelerated speed, which becomes uniform when the forces produced by the radiation are balanced by the friction of the pivot and of the residual air in the globe.

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  • The name radiometer arose from an idea that the final steady speed of rotation might be utilized as a rough measure of the intensity of the exciting radiation.

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  • Rotation is communi cated by a pinion, turned by the handle c (concealed in the figure), which works in teeth cut on the edge of the flange h.

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  • The counterpoise w balances the head about its axis of rotation.

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  • The ring c, which carries the supports of the handles a', b', is capable of a certain amount of rotation on the tube.

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  • Complete rotation of the head is obviously impossible because of the interference of the declination axis with the rods, and therefore, in some angles, objects cannot be measured in two positions of the circle.

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  • It permits complete rotation of the tube and measurement of all angles in reversed positions of the circle; the handles that move the slides can be brought down to the eye-end, inside the tube, and consequently made to rotate with it; and the position circle may be placed at the end of the cradle next the eyeend where it is convenient of access.

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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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  • p. 419) suggested the division of the small speculum of a Cassegrain telescope and the production of double image by micrometric rotation of the semispecula in the plane passing through their axis.

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  • In a direction perpendicular to that axis it must be so placed that when it is moved by rotation of the axis bb the separation of the images shall be parallel to that motion.

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  • Hence the geoid or figure of the sea-surface is not part of an ellipsoid of rotation but is irregular.

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  • Further, all water particles when moving undergo a deviation from a straight path due to the forces set up by the rotation of the earth deflecting them towards the right as they move in the northern hemisphere and towards the left in the southern.

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  • The current set up by the grip of the wind sweeping over the surface is deflected by the earth's rotation about 45° to the right of the direction of the wind in the northern hemisphere and to the left in the southern.

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  • When a drift-current impinges directly upon a coast there is a heaping up of surface water, giving rise to a counter-current in the depths, which maintains the level, and this counter-current, although subject to deflection on account of the rotation of the earth, is deflected much less than a pure drift-current would be.

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  • The actual direction of this circulation is strongly modified by the influence of the earth's rotation.

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  • Both outflowing and inflowing currents are subject to the deflection towards the right imposed by the earth's rotation.

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  • The tub when brought to the surface, after passing over a weigh-bridge where it is weighed and tallied by a weigher specially appointed for the purpose by the men and the owner jointly, is run into a " tippler," a cage turning about a horizontal axis which discharges the load in the first half of the rotation and brings the tub back to the original position in the second.

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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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  • Thus when n = o, the whole energy must be translational: there can be no energy of rotation or of internal motion.

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  • The molecules of gases for which n = o must accordingly be spherical in shape and in internal structure, or at least must behave at collisions as though they were spherical, for they would otherwise be set into rotation by the forces experienced at collisions.

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  • The value n = 2 is appropriate to bodies of which the shape is that of a solid of revolution, so that there is no rotation about the axis of symmetry.

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  • Buonanni, a learned Jesuit of Rome) that they are not generated out of the mud or sand found on the seashore or the beds of rivers at low water, but from spawn, by the regular course of generation; and he maintained the same to be true of the fresh-water mussel (Unio), whose ova he examined so carefully that he saw in them the rotation of the embryo, a phenomenon supposed to have been first discovered long afterwards.

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  • formulae, not involving the conception of an angle as generated by rotation, belong to this stage; the additional geometrical idea involved being that of the proportionality of the sides of similar triangles.

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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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  • The treatment of an angle as generated by rotation, the investigation of the relations between trigonometrical ratios and circular measure, the application of interpolation to trigonometrical tables, and the general use of graphical methods to represent continuous variation, all imply an analytical onlook, and must therefore be deferred to this stage.

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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 elementary conception is of the figure as already existing, rather than of its method of description; the test of circularity being the possibility of rotation within a surrounding figure so as to keep the two boundaries always completely in contact.

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  • The new constitution provided for a boule whose members were to be recruited by lot from all citizens above thirty; the functions of this body to be exercised by four sections succeeding one another by yearly rotation and serving without pay; all high officials to be chosen by it out of its own members.

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  • The pitch of this note will rise as the rate of rotation increases,_and becomes steady when that rotation is maintained uniform.

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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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  • 16), and will therefore urge the latter to rotation round its centre.

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  • After 1/nth of a revolution, the two sets of perforations will again coincide, the lateral impulse of the air repeated, and hence the rapidity of rotation increased.

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  • This will go on continually as long as air is supplied to the cylinder, and the velocity of rotation of the upper plate will be accelerated up to a certain maximum, at which it may be maintained by keeping the force of the current constant.

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  • The result is a note whose pitch rises as the velocity of rotation increases, and becomes steady when that velocity reaches its constant value.

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  • If the wheel has very little work to do it may not be necessary to apply driving power, and uniform rotation may be maintained by the electro-magnet.

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  • The arc of rotation is 82°, and the axis of rotation is 13 ft.

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  • By rot ting a small electro-magnet in water, between the poles of ano her magnet, and then measuring the heat developed in the wat r and other parts of the machine, the current induced in the coils, and the energy required to maintain rotation, he cal b lated that the quantity of heat capable of warming one you d of water one degree F.

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  • The rotation of the mercury is detected and measured by means of a small vane of platinum wire immersed in it, the shaft of this vane being connected by an endless screw with a counting mechanism.

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  • The driving force is balanced against a retarding force produced by the rotation of a copper disk fixed on the armature shaft, which rotates between the poles of a permanent magnet.

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  • Induced or eddy currents are thus created in the copper disk, and the reaction of these against the magnetic field offers a resistance to the rotation of the disk.

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  • The various members of the body were parcelled out among the nakshatras, and a rotation of food was prescribed as a wholesome accompaniment of the moon's revolution among them.8 1 Max Muller, op. cit., p. lxiv.

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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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  • Lord Kelvin was thereby induced to identify magnetic force with rotation, involving, therefore, angular momentum in the aether.

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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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  • 184, 254) also deserve mention, as well as his discussions of the rotation of Mars, by which he deduced its period with a probable error of 0 9.005.

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  • The speculum lies face upwards in a shallow bath of water (to preserve a uniform temperature), and the polisher fits loosely in a ring, so that the rotation of the speculum makes it revolve also, but more slowly.

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  • It is evidently a gyroscopic effect, being reversed in direction by a change from a right to a left-handed twist of rifling, and being increased by an increase of rotation of the shot.

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  • After a certain discount for friction and the recoil of the gun, the net work realized by the powder-gas as the shot advances AM is represented by the area Acpm, and this is equated to the kinetic energy e of the shot, in foot-tons, (I) e d2 I + p, a in which the factor 4(k 2 /d 2)tan 2 S represents the fraction due to the rotation of the shot, of diameter d and axial radius of gyration k, and S represents the angle of the rifling; this factor may be ignored in the subsequent calculations as small, less than I %.

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  • In response to the demand for self-government, in September 1647 he and the council appointed - after the manner then followed in Holland - from eighteen representatives chosen by the people a board of nine to confer with him and the council whenever he thought it expedient to ask their advice; three of the nine, selected in rotation, were permitted to sit with the council during the trial of civil cases; and six were to retire each year, their successors to be chosen by the director and council from twelve candidates nominated by the board.

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  • The Mongolian peoples not only count their lunar months by these signs, but they reckon the successive days by them, rat-day, bull-day, tiger-day, &c., and also, by combining the twelve signs in rotation with the elements, they obtain a means of marking each year in the sixty-year cycle, as the woodrat year, the fire-tiger year, &c. This method is highly artificial, and the reappearance of its principle in the Mexican and Central American calendar is suggestive of importation from Asia.

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  • The cycle of 52 years was reckoned by combining these signs in rotation with numbers up to 13, thus: I rabbit, 2 reed, 3 flint, 4 house, 5 rabbit, 6 reed, &c. By accident this calendar may be exactly illustrated with a modern pack of cards laid out in rotation of the four suits, as, ace of hearts, 2 of spades, 3 of diamonds, 4 of clubs, 5 of hearts, 6 of spades, &c. In the Mexican ritual calendar of the days of the year, the same method is carried further, the series of twenty day-signs being combined in rotation with numbers up to 13; as this cycle of days only reaches 260, a series of nine other signs are affixed in addition, to make up the 365-day year.

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  • It is plain that this rotation of signs served no, useful purpose whatever, being less convenient than ordinary counting such as the Mexicans employed in their other calendar already mentioned, where the 20-day periods had each a name like our' months, and their days had signs in regular order.

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  • in rotation.

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  • Among his happy conjectures may be mentioned that of the sun's axial rotation, postulated by him as the physical cause of the revolutions of the planets, and soon after confirmed by the discovery of sun-spots; the suggestion of a periodical variation in the obliquity of the ecliptic; and the explanation as a solar atmospheric effect of the radiance observed to surround the totally eclipsed sun.

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  • The first of these features is determined by the intermediate position of the United States between the equator and the north pole; the second by the equatorial-polar temperature contrast and the eastward rotation of the planet.

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  • Consequently there were large tracts of untilled " waste " land; but these rapidly responded to fertilization and rotation of crops, often yielding Boo to 1200 lb of cotton per acre, and Georgia in 1899 used more fertilizers than any other state in the Union.

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  • There are 1 Owing to the custom which holds in Georgia of choosing state senators in rotation from each of the counties making up a senatorial district, it happened in 1907 that few cities were represented directly by senators chosen from municipalities.

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  • In common with all other Coelomata, the Mollusca are at one period of life possessed of a prostomium or region in front of the mouth, which is the essential portion of the " head," and is connected with the property of forward locomotion in a definite direction and the steady carriage of the body (as opposed to rotation of the body on its long axis).

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  • No special rotation is followed: indeed the soil best suited for rice is ill adapted for any other crop. In some cases little manure is employed, but in others abundance of manure is used.

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  • But frequent exceptions have been observed in which the direction of rotation is reversed.

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  • Secondly, his theory of inference contains the admission that we infer beyond sensations: he remarks that the space of the geometer is beyond space-sensations, and the time of the physicist does not coincide with time-sensations, because it uses measurements such as the rotation of the earth and the vibrations of the pendulum.

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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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  • His first notable discovery was the production of the continuous rotation of magnets and of wires conducting the electric current round each other.

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  • Faraday was not there at the time, but coming in afterwards he heard the conversation on the expected rotation of the wire.

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  • This led him in the beginning of September to discover the method of producing the continuous rotation of the wire round the magnet, and of the magnet round the wire.

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  • Ampere, Wollaston and others, the realization of the continuous rotation of the wire and the magnet round each other was a scientific puzzle requiring no mean ingenuity for its original solution.

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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 negative rotation in ferro-magnetic media is the only fact of importance which remained to be discovered afterwards (by M.

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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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  • The results thus obtained apply to the motion of a small body, the rotation of which is disregarded.

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  • He attained correct views as to the character of centrifugal force in connexion with Galileo's theory; and, when the fact of the variation of gravity (Galileo's acceleration) in different latitudes first became known from the results of pendulum experiments, he at once perceived the possibility of connecting such a variation with the fact of the earth's diurnal rotation relatively to the stars.

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  • This is the earliest example of a theoretical investigation of the rotation of rigid bodies.

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  • This test involves only changes of velocity, and so does not distinguish between two bases, each of which moves relatively to the other with uniform velocity without rotation.

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  • In the treatment of the relative motions of a limited system, we may use a confessedly provisional base, though it may be necessary to introduce corrections, either exact or approximate, to take account either of the existence of bodies outside the system, or of the rotation of the base employed relative to a more correct one.

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  • Differences of acceleration due to the attractions of the sun and moon are not important for terrestrial systems on a small scale, and can usually be ignored, but their effect (in combination with the rotation of the earth) is very apparent in the case of the ocean tides.

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  • A more considerable defect is due to the earth having a diurnal rotation relative to a Newtonian base, and this is never wholly ignored.

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  • Take a base attached to the centre of the earth, but without this diurnal rotation.

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  • Another way of stating the same thing is to say that we introduce, as a correction for the earth's rotation, a force called "centrifugal force," which combined with gravitation gives the weight of the body.

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  • In the case of a body moving relatively tothe earth, the introduction of centrifugal force only partially corrects the effect of the earth's rotation.

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  • No correction for any defect in it has been found necessary; moreover, no rotation of the base relative to the directions of the stars without proper motion has been detected.

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  • Practically clocks are regulated by reference to the diurnal rotation of the earth relatively to the stars, which affords a measurement on the repetition principle agreeing with other methods, but more accurate than that given by any existing clock.

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  • 4, which, according to the kinetic theory, is an indication that an important fraction of the energy absorbed is devoted to rotation or vibration.

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  • Sigismund died very suddenly in his 66th year, leaving two sons, Wladislaus and John Casimir, who succeeded him in rotation.

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  • Any who expect to get early cabbage, cauliflower, lettuce or radishes, while planting or sowing is delayed until the time of sowing tomato and egg plant in May, are sure to be disappointed of a full crop. Frequent rotation of crops should be practised in the vegetable garden, in order to head off insects and diseases; and also to make the best use of the land.

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  • If this series of operations be made to depend upon the continuous rotation of a winch or handle, the arrangement constitutes an electrostatic influenceenachine.

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  • The rotation, by destroying the contacts, preserves this unequal distribution, and carries B from A to C at the same time that the tail K connects the ball with the plate C. In this situation, the electricity in B acts upon that in C, and produces the contrary state, by virtue of the communication between C and the ball; which last must therefore acquire an electricity of the same kind with that of the revolving plate.

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  • But the rotation again destroys the contact and restores B to its first situation opposite A.

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  • A second rotation will, of course, produce a proportional augmentation of these increased quantities; and a continuance of turning will soon bring the intensities to their maximum, which is limited by an explosion between the plates" (Phil.

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  • The rotation continuing the ball thus negatively charged is made to give up this charge to that negatively electrified field plate, and the ball positively charged its charge to the positively electrified field plate, by touching little contact springs.

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  • In the course of their rotation two diametrically opposite carriers touched against the ends of a neutralizing conductor so as to form for a moment one conductor, and the moment afterwards these two carriers were insulated, one carrying away a positive charge and the other a negative.

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  • Continuing their rotation, the positively charged carrier gave up its positive charge by touching a little knob attached to the positive field plate, and similarly for the negative charge carrier.

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  • The positive stud on the back plate will act inductively on the front stud and charge it negatively, and similarly for the other stud, and as the rotation continues these charged studs will pass round and give up most of their charge through the combs to the Leyden jars.

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  • The combination of ductility, which lessens the tendency to break when overstrained or distorted, with a very high limit of elasticity, gives it great value for shafting, the merit of which is measured by its endurance of the repeated stresses to which its rotation exposes it whenever its alignment is not mathematically straight.

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  • The extensive cultivation of beetroot, of potatoes for distilleries, and of fodder crops has led to the introduction of a rotation of several years instead of the former " three-fields " system; and agricultural machinery is in more general use, especially on the larger estates of the west.

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  • This encourages a rotation of crops and enables the precious liquid to be carried over a larger area than could be done otherwise.

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  • The three years rotation formerly in use, where autumn and spring-sown grain and fallow succeeded each other, has now been abandoned, except in some districts, where the system has been modified and improved.

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  • The general public were content to find the explanation of the movements in spirits, animal magnetism, odic force, galvanism, electricity, or even the rotation of the earth.

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  • In some cases changes of intensity take place round the auroral zenith, simulating the effect that would be produced by a cyclonic rotation of luminous matter.

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  • By this rotation a beam of light reflected from the surface suffers displacement.

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  • The necks of vases were the first part finished with rotation, at the middle and close of the prehistoric age.

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  • In the working of this type of kiln the rotation and slight inclination of the cylinder cause the raw material to descend towards the lower end.

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  • The material continues to descend by the rotation of the kiln and reaches the lower end nearest ?

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  • In the Theorie nouvelle de la rotation des corps (1834) he treats the motion of a 'rigid body geometrically, and shows that the most general motion of such a body can be represented at any instant by a rotation about an axis combined with a translation parallel to this axis, and that any motion of a body of which one point is fixed may be produced by the rolling of a cone fixed in the body on a cone fixed in space.

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  • Atwood's published works, exclusive of papers contributed to the Philosophical Transactions, for one of which he obtained the Copley medal, are as follows: - Analysis of a Course of Lectures on the Principles of Natural Philosophy (Cambridge, 1784); Treatise on the Rectilinear Motion and Rotation of Bodies (Cambridge, 1784), which gives some interesting experiments, by means of which mechanical truths can be ocularly exhibited and demonstrated, and describes the machine, since called by Atwood's name, for verifying experimentally the laws of simple acceleration of motion; Review of the Statutes and Ordinances of Assize which have been established in England from the 4th year of King John, 1202, to the 37th of his present Majesty (London, 1801), a work of some historical research; Dissertation on the Construction and Properties of Arches (London, 1801), with supplement, pt.

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  • In 1905 the yield of hay from clover, sainfoin and rotation grasses amounted to 666,985, tons, or 31.19 cwts.

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  • Most noteworthy, however, in this connexion is the fact that he anticipated Copernicus by maintaining the theory of the rotation of the earth.

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  • The axis of rotation AB bears a rigidly attached rod DBC inclined to it at an angle equal to the sun's polar distance.

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  • AB is the axis of rotation, BC an adjustable FIG.

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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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  • In Se p tember of that year he discovered that the force required for the rotation of a copper disk becomes greater when it is made to rotate with its rim between the poles of a magnet, the disk at the same time becoming heated by the eddy or "Foucault currents" induced in its metal.

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  • He studied mathematics and physics in his native town, Groningen, where in 1879 he took his doctor's degree on presenting a dissertation entitled New Proofs of the Earth's Rotation.

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  • One necessary effect of the tidal retardation of the earth's rotation is gradually to increase this length.

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  • As this amount seems to be markedly less than that which would be expected from the cause in question, it is probable that some other cause tends to accelerate the earth's rotation and so to shorten the day.

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  • The moon's apparent mean motion in longitude seems also to indicate slow periodic changes in the earth's rotation; but these are not confirmed by transits of Mercury, which ought also to indicate them.

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  • The rotation of crops is not adopted as a principle of cultivation; but in practice it is well known that a succession of exhausting crops cannot be taken in consecutive seasons from the same field, and the advantage of fallows is widely recognized.

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  • He demonstrated the rotation of the satellites of Jupiter round the planet, and gave rough predictions of their configurations, proved the rotation of the sun on its axis, established the general truth of the Copernican system as compared with that of Ptolemy, and fairly routed the fanciful dogmas of the philosophers.

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  • Thus, when the declination axis is horizontal the telescope moves in the plane of the meridian by rotation on the declination axis only.

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  • Thus one important attribute of an equatorially mounted telescope that, if it is directed to any fixed star, it will follow the diurnal motion of that star from rising to setting by rotation of the polar axis only.

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  • The rotation of the dome, and p _z with it the platform-framework, is accomplished s by means of electric motors, as also is the raising and lowering of the platform on its = =_ framework.

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  • There are farther inconveniences in the use of such a telescope, viz., that the image undergoes a diurnal rotation about the axis of the horizontal telescope, so that, unless the sensitive plate is also rotated by clockwork, it is impossible to obtain sharp photographs with any but instantaneous exposures.

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  • On this principle the use of the level is abolished, the telescope is mounted on a metallic float, and it is assumed that, in course of the rotation of this float, the zenith distance of the axis of the telescope will remain undisturbed, that is, of course, after the undulations, induced by the disturbance of the mercury, have ceased.

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  • For Diatomic Or Compound Gases Clerk Maxwell Supposed That The Molecule Would Also Possess Energy Of Rotation, And Endeavoured To Prove That In This Case The Energy Would Be Equally Divided Between The Six Degrees Of Freedom, Three Of Translation And Three Of Rotation, If The Molecule Were Regarded As A Rigid Body Incapable Of Vibration Energy.

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  • Boltzmann Suggested That A Diatomic Molecule Regarded As A Rigid Dumb Bell Or Figure Of Rotation, Might Have Only Five Effective Degrees Of Freedom, Since The Energy Of Rotation About The Axis Of Symmetry Could Not Be Altered By Collisions Between The Molecules.

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  • Since Much Smaller Values Are Found For More Complex Molecules, We May Suppose That, In These Cases, The Energy Of Rotation Of A Polyatomic Molecule May Be Greater Than Its Energy Of Translation, Or Else That Heat Is Expended In Splitting Up Molecular Aggregates, And Increasing Energy Of Vibration.

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  • It Is Not At All Clear, However, That Energy Of Vibration Should Bear A Constant Ratio To That Of Translation, Although This Would Probably Be The Case For Rotation.

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  • For Such Gases, Assuming A Constant Ratio Of Rotation To Translation, The Specific Heat At Low Pressures Would Be Very Nearly Constant.

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  • "If a body moves in any direction on the earth's surface, there is a deflecting force arising from the earth's rotation, which deflects it to the right in the northern hemisphere and to the left in the southern hemisphere."

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  • Thus rotation is now expressed in symbols at least as simply as it can be exhibited by means of a model.

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  • Twinning according to the first law is effected by rotation about an axis normal to the sphenoidal face (III), the resulting form resembling the twins of blende and spinel.

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  • Twinning according to the second law can only be explained by reflection across the plane (roi), not by rotation about an axis; chalcopyrite affords an excellent example of this comparatively rare type of symmetric twinning.

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  • But his name is chiefly perpetuated through his investigation of the motions of sun-spots, by which he determined the elements of the sun's rotation and made the important discovery of a systematic drift of the photosphere, causing the rotation-periods of spots to lengthen with increase of solar latitude.

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  • Since the movements of the magnet are always small, the rotation of the magnet is proportional to the change in H, so long as M and the couple, 0, corresponding to unit twist of the suspension system remain constant.

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  • By this arrangement the angular rotation of the reflected beam is less than that of the magnet, and hence the spot of light reflected from this mirror yields a trace on a much smaller scale than that given by the ordinary mirror and serves to give a complete record of even the most energetic disturbance.

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  • Chasles any displacement whatever of the lamina in its own plane is equivalent to a rotation about some finite or infinitely distant point J.

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  • For suppose that in consequence of the displacement a point of the lamina is brought from A to B, whilst the point of the lamina which was originally at B is brought to C. Since AB, BC, are two different positions of the same line in the B C lamina they are equal, and it is evident that the rotation could have been effected by a rotation about J,

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  • The latter may be brought from any one of its positions to a neighboring one by a rotation about the proper centre.

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  • If P, P be consecutive positions of the same point, and the corresponding angle of rotation, then ultimately PP is at right angles to JP and equal to JP. ~O.

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  • successive centres of rotation.

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  • If now the assemblage be brought into any other position relative to the earth, without alteration of the mutual distances, this is equivalent to a rotation of the directions of the forces relatively to the assemblage, the ratios of the forces remaining unaltered.

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  • It was shown by Euler (1776) that any displacement in which one point 0 of the body is fixed is equivalent to a pure, rotation about some axis through 0.

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  • The spherical A isosceles triangles AJB, BJC are con gruent, and we see that AB can be brought into the position BC by a rotation about the axis OJ through an FIG.

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  • It is convenient to distinguish the two senses in which rotation may take place about an axis OA by opposite signs.

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  • We shall reckon a rotation as positive when it is related to the direction from 0 to A as the direction of rotation is related to that of translation in a right-handed screw.

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  • Thus a, negative rotation about OA may be regarded as a positive rotation about OA, the prolongation of AO.

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  • Now suppose that a body receives first a positive rotation a about OA, and secondly a positive rotation e3 about OB; and let A, B be the intersections of these axes with a sphere described about 0 as centre.

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  • To find the angle of the equivalent rotation, in the actual case, suppose that the second rotation (about OB) brings a point from A to A.

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  • 39) are symmetrically equal, and the angle of the resultant rotation, viz.

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  • In any continuous motion of a solid about a fixed point 0, the limiting position of the axis of the rotation by which the body can be brought from any one of its positions to a consecutive one is called the instantaneous axis.

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  • It follows from Eulers theorem that the most general displacement of a rigid body may be effected by a pure translation which brings any one point of it to its final position 0, followed by a pure rotation about some axis through 0.

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  • Hence, if ~, tr denote the initial and final positions of any figure in one of these planes, the displacement could evidently have been effected by, (I) a translation perpendicular to the planes in question, bringing ointo some position o- in the plane of a., and (2) a rotation about a normal to the planes, bringing a.

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  • In other words, the most general displacement is equivalent to a translation parallel to a certain axis combined with a rotation about that axis; i.e.

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  • In particular cases, of course, the translation, or the rotation, may vanish.

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  • This is the axis of the required screw; the amount of the translation is measured by the projection of AB or BC or CD on the axis; and the angle of rotation is given, by the inclination of the aforesaid bisectors.

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  • a rotation through two right angles) as the fundamental operation.

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  • This has the advantage that it is completely specified by the axis of the rotation; the sense being immaterial.

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  • Successive halfturns about intersecting axes a, b are equivalent to a rotation about the common perpendicular to a, b at their intersection, Of amount equal to twice the acute angle between them, in the direction from a to b.

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  • Successive half-turns about two skew axes a, b are equivalent to a twist about a screw whose axis is the common perpendicular to a, b, the translation being double the shortest distance, and the angle of rotation being twice the acute angle between a, b, in the direction from a to b.

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  • An infinitely small rotation about any axis is conveniently represented geometrically by a length AB measures along the axis and proportional to the angle of rotation, with the convention that the direction from A to B shall be related to the rotation as is the direction of translation to that of rotation in a righthanded screw.

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  • rotation a+/3 about a parallel ---- --

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  • It thus appears that an infinitesimal rotation is of the nature of a localized vector, and is subject in all respects to the same mathematical Jaws as a force, conceived as acting on a rigid body.

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  • the positive directions of the axes are assumed to be so arranged that a positive rotation of 90 about Ox would bring Oy into the position of UI, and so on.

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  • The displacement will consist of an infinitesimal rotation e about some axis through U, whose direction-cosines are, say, 1, m, n.

    0
    0
  • From the equivalence of a small rotation to a localized vector it follows that the rotation ~ will be equivalent to rotations E,ii, ~ about Ox, Oy, Uz, respectively, provided = le, s1 = me, i nc (I) and we note that li+,72+l~Z~i (2)

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  • The points whose displacements are in the direction of the resultant axis of rotation are determined by Ix: Iy: Iz = E: ~: ~, or (X +nz ly)IE = Cu + Ix ~z)/v = (e +:~y ~x)ft.

    0
    0
  • The translation parallel to this axis is lox + mly + nhz (Xf + uv + vi) Ic. (8) The linear magnitude which measures the ratio of translation to rotation in a screw is called the pitch.

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    0
  • When the latter invariant, but not the former, vanishes, the displacement is equivalent to a pure rotation.

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    0
  • 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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  • 47) in the plane of the paper, and let J be the point where this plane is met by the axis of rotation.

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    0
  • Draw JBA perpendicular to the lines of action, and let e be the angle of rotation.

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    0
  • If the twist is a pure rotation, these quantities are subject to the relation A~+u,i+w~=0.

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    0
  • If a line is a null-line with respect to the wrench (X, Y, Z, L, M, N), the work done in an infinitely small rotation about it is zero, and its coordinates are accordingly subject to the further relation Lf+M~+Nl+XX+Yp+Zvo, (5~

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  • For by giving the body (in imagination) a displacement of translation we learn that the sum of the resolved parts of the forces in any assigned direction is zero, and by giving it a displacement of pure rotation we learn that the sum of the moments about any assigned axis is zero.

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  • It is found that for almost all purposes a system of measurement based ultimately on the earths rotation is perfectly adequate.

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  • Two-dimensional~ Problems.In the case of rotation about a fixed axis, the principles take a very simple form.

    0
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  • d/dI.(Mu) =X; it shows that I measures the inertia of the body as regards rotation, just as M measures its inertia as regards translation.

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    0
  • As a final example we may note the arrangement, often employed in physical measurements, where a body performs small oscillations about a vertical axis through its mass-centre G, under the influence of a couple whose moment varies as the angle of rotation from the equilibrium position.

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    0
  • Take rectangular axes, of which Oz coincides with the axis of rotation.

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    0
  • unless the axis of rotation be a principal axis of inertia (ii) at 0.

    0
    0
  • the mass-centre must lie in the axis of rotation.

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    0
  • It is to be carefully noticed that the axis of resultant angular momentum about 0 does not in general coincide with the instantaneous axis of rotation.

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    0
  • It appears / that if the body be sightly dis turbed from a state of rotation about the principal axis of / greatest or least moment, the invariable cone will closely sur round this axis, which will therefore never deviate far - from the invariable line.

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    0
  • If, on the other hand, the body be slightly disturbed from a state F 8 of rotation about the mean axis 1G.

    0
    0
  • Hence a rotation about the axis of greatest or least moment is reckoned as stable, a rotation about the mean axis as unstable.

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  • Hence if the earths axis of rotation deviates slightly from the axis of figure, it should describe a cone about the latter in 320 sidereal days.

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    0
  • The absence of g from the latter expressior indicates that the circumstances of the rapid precession are verb nearly those of a free Eulerian rotation (~ 19), gravity playing only a subordinate part.

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  • This is, in fact, the invariable line of the free Eulerian rotation with which (as already remarked) we are here virtually concerned.

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    0
  • If in (2) we put, L, M, N=O we get the case of free rotation; thus A~f=(BC)gr, B~=(CA)rp, (5)

    0
    0
  • These equations are due to Euler, with whom the conception of moving axes, and the application to the problem of free rotation, originated.

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  • Since 8, o, are the polar co-ordinates (in a horizontal plane) of a point on the axis of symmetry, relative to an initial line which revolves with constant angular velocity P/2A, we see by comparison with 14 (i5) (16) that the motion of such a point will be elliptic-harmonic superposed on a uniform rotation e/2A, provided v2 >4AMgh.

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  • If in (21) we imagine that x, y, I denote infinitesimal rotations of a solid free to turn about a fixed point in a given field of force, it appears that the three normal modes consist each of a rotation about one of the three diameters aforesaid, and that the values of in are proportional to the ratios of the lengths of corresponding diameters of the two quadrics.

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  • Poinsot, Elements de statique (Paris, 1804), and Th-orie nouvelle de la rotation des corps (Paris, 1834).

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  • This being the case, the various possible motions of a rigid solid body may all be classed under the following heads: (I) Shifting or Translation; (2) Turning or Rotation; (3) Motions compounded of Shifting and Turning.

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  • Rotation about a Fixed Axis: Lever, Wheel and Axle.The fixed axis of a turning body is a line fixed relatively to the body and relatively to the fixed space in which the body turns.

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  • Let the angular velocity of the rotation be denoted by a=dO/dt, then the linear velocity of any point A at the distance r from the axis is or; and the path of that point is a circle of the radius r described about the axis.

    0
    0
  • in any direction AP parallel to the D plane of rotation is equal to the total velocity of the point m, found by letting fall Fm perpendicular to AP;

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    0
  • 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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  • 91), when the rolling will be towards the same hand as th rotation, or inside (as in fig.

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  • To find, then, the direction and velocity at the given instant of any point P, either in or rigidly attached to the rolling cylinder T, draw the plane PT; the direction of motion of P will be perpendicular to that plane, and towards the right or left hand according to the direction of the rotation of bbb; and the velocity of P will be v~=y.PT, (3)

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  • 92, AB is to be treated as negative, which will give a negative value to a, indicating that in this case the rotation of AB round A is contrary to that of the cylinder bbb.

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  • By reasoning similar to that of 30, it appears that OT is the instantaneous axis of rotation 01 the rolling cone.

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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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    0
  • 32~ Screw-like or Helical Motson.Since any displacement in a plane can berepresented in general by a rotation, it follows that the only combination of translation and rotation, in which a complex movement which is not a mere rotation is produced, occurs when there is a translation perpendicular to the plane and parallel to the axis of rotation.

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  • Such a complex motion is called screw-like or helical motion; for each point in the body describes a helix or screw round the axis of rotation, fixed or instantaneous as the case may ~ be.

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  • Helical motion and screws adapted to it are said to be right- or left-handed according to the G appearance presented by the rotation to an observer looking towards the direction of the translation.

    0
    0
  • Let V5 denote the velocity of advance at a given instant, which of course is common to all the particles of the body; a the angular velocity of the rotation at the same instant; 2,r = 6.2832 nearly, the circumference of a circle of the radius unity.

    0
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  • is the p-itch or o4va~1ce per turna length which expreases the comparative motion of the translation and the rotation.

    0
    0
  • Application to a Pair of TurnIng Fseces.Let ai, a2 be the angular velocities of a pair of turning pieces; Of, Oi the angles which their line of connection makes with their respective planes of rotation; Ti, r2 the common perpendiculars let fall from the line of connection upon the respective axes of rotation of the pieces.

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  • Application to a Shifting Piece and a Turning Piece.Let a shifting piece be connected with a turning piece, and at a given instant let ai be the angular velocity of the turning piece, r~ the common perpendicular of its axis of rotation and the line of connection, Oi the angle made by the line of connection with the plane of rotation, Oi the angle made by the line of connection with the direction of motion 01 the shifting piece, v2 the linear velocity -of that piece.

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  • If the velocity ratio is to be variable, as in s Williss Class B, the figures of the wheels are a pair of rolling curves, subject to the condition that the distance between their poles (which are the centres of rotation) shall be constant.

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  • To illustrate this subject, it may be mentioned that an ellipse rotating about one focus rolls completely round in outside gearing with an equal and similar ellipse also rotating about one focus, the distance between the axes of rotation being equal to the major axis of the ellipses, and the velocity ratio varying from to I ~eccentricitY an hyperbola rotating about its further focus rolls in inside gearing, through a limited arc, with an equal and similar hyperbola rotating about its nearer focus, the distance between the axes of rotation being equal to the axis of the hypereccentricity + I

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  • Sliding Cont?,zct (lateral): Skew-Bevel Wheels.An hyperboloid of revolution is a surface resembling a sheaf or a dice box, generated by the rotation of a straight iAn line round an axis from which it is at a constant distance, - c and to which it is inclined at -~- a constant angle.

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  • The directions and positions of the axes being given, and the required angular velocity ratio, the following construction serves to determine the line of contact, by whose rotation round the two axes respectively the hyperboloids are generated: In fig.

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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.

    0
    0
  • Thus the relative motion of the wheels is unchanged; but I is considered as fixed, and 2 has the total motion, that is, a rotation about the instantaneous axis I, with the angular velocity cii+a1.

    0
    0
  • Hence the velocity of sliding is that due to this rotation about I, with the radius IT; that is to say, its value is (ai+ai).IT; (26)

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  • The relation between the advance and the rotation, which compose the motion of a screw working in contact with a fixed screw or helical guide, has already been demonstrated in 32; and the same relation exists between the magnitudes of the rotation of a screw about a fixed axis and the advance of a shifting nut in which it rotates.

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  • The angular velocities of the two disks and of the cross are all equal at every instant; the middle point of the cross, at A, revolves in the dotted circle described upon the line of centres C1C, as a diameter twice for each turn of the disks and cross; the instantaneous axis of rotation of the cross at any instant is at I, the point in the circle Cf C2 diametrically opposite to A.

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  • 108, A, reverses the direction of the rotation communicated; an uncrossed belt, as in fig.

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  • In a turning piece, the perpendicular let fall from its connected point upon its axis of rotation is the arm or crank-arm.

    0
    0
  • The axes of rotation of a pair of turning pieces connected by a link are almost always parallel, and perpendicular to the line of connection n which case the angular velocity ratio at any instant is the recipocal of the ratio of the common perpendiculars let fall from the me of connection upon the respective axes of rotation.

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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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  • Supposing all the wheels to be in outside gearing, then, as each elementary combination reverses the direction of rotation, and as the number of elementary combinations m 1 is one less than the number of axes rn it is evident that if m is odd the direction of rotation is preserved, and if even reversed.

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  • Method 1.By reference to 30 it will be seen that the motion of a cylinder rolling on a fixed cylinder is one of rotation about an instantaneous axis T, and that the velocity both as regards direction and magnitude is the same as if the rolling piece B were for the instant turning about a fixed axis coincident with the instantaneous axis.

    0
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  • A couple is said to be right or left handed with reference to the observer, according to the direction in which it tends to turn the body, and is a driving couple or a resisting couple according as its tendency is with or against that of the actual rotation.

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  • When the relative motion of the rubbing surfaces is one of rotation, the work of friction in a unit of time, for a portion of the rubbing surfaces at a given distance from the axis of rotation, may be found by multiplying together the friction of that portion, its distance from the axis, and the angular velocity.

    0
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  • The product of the force of friction by the distance at which it acts from the axis of rotation is called the moment of friction.

    0
    0
  • To express this symbolically, let dii represent the area of a portion of a pair of rubbing surfaces at a distance r from the axis of their relative rotation; p the intensity of the normal pressure at du per unit of area; and f the coefficient of friction.

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    0
  • The side on which it touches that circle B C is determined by the fact that the obliquity A of the reaction is such as to oppose the A rotation.

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  • When the clutch is moved towards the hoop, its arms catch those of the hoop, and cause the hoop to rotate and to communicate its rotation to the pulley by friction.

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  • It is of the nature of a couple, resisting rotation.

    0
    0
  • 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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  • If the axis of rotation traverses the centre of gravity of the body, the centrifugal force exerted on that axis is nothing.

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  • Hence, unless there be some reason to the contrary, each piece of a machine should be balanced on its axis of rotation; otherwise the centrifugal force will cause strains, vibration and increased friction, and a tendency of the shafts to jump out of their bearings.

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  • Centrifugal Couples of a Rotating Body.Besides the tendency (if any) of the combined centrifugal forces of the particles of a rotating body to s/lift the axis of rotation, they may also tend to turn it out of its original direction.

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  • The latter tendency is called a centrifugal couple, and vanishes for rotation about a principal axis.

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  • It is essential to the steady motion of every rapidly rotating piece in a machine that its axis of rotation should not merely traverse its centre of gravity, but should be a permanent axis; for otherwise the centrifugal couples will increass friction, produce oscillation of the shaft and tend to make it leave its bearings.

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  • * Method of computing the position and magnitudes of balance weights which must be added to a given system of arbitrarily chosen rotating masses in order to make the common axis of rotation a permanent axis.The method here briefly explained, is taken from a paper by W.

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  • Hence finally the conditions which must be satisfied by the system of weights in order that the axis of rotation may be a permanent axis is (I) (Wiri+W2r2+Wsr,)=o (2) (~Vja1r1+\Via1ri+)=o (c)

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  • To take a simple case, suppose a shaft supported on two bearings to carry a disk of weight W at its centre, I and let the centre of gravity of the disk be at a distance e from the axis of rotation, this small distance being due to imperfections of material or faulty construction.

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  • Neglecting the mass of the shaft itself, when the shaft rotates with an angular velocity a, the centrifugal force Wae/g will act upon the shaft and cause its axis to deflect from the axis of rotation a distance, y say.

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  • The actual energy due to the rotation of the fly, with its mean angular velocity, is equal to one-half of the periodical excess of energy multiplied by the steadiness.

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  • 133) let the required deviation be a rotation of the bod3 BB about an axis 0, not traversing the centre of gravity G, di being, as before, the deviation of angular velocity to be produced in the interval dl.

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  • A rotation with the angular velocity a about an axis 0 may be considered as compounded ~ of a rotation with the same angular velocity about an axis drawn through G parallel to 0

    0
    0
  • and a deviation da of rotation about an Fl G.

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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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  • In the midst of a burden of letterwriting, the minute details in his diaries of tree-planting and rotation of crops, and his increasing reading on the political side of history, he found time to entertain a stream of visitors from all parts of the United States and from abroad.

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  • In 1821 Michael Faraday (1791-1867), who was destined later on to do so much for the science of electricity, discovered electromagnetic rotation, having succeeded in causing a wire conveying a voltaic current to rotate continuously round the pole of a permanent magnet.

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  • The study of the relation between the magnet and the circuit conveying an electric current then led Arago to the discovery of the " magnetism of rotation."

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  • Much ingenuity had, however, to be expended before a method was found of exhibiting such a rotation.

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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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  • In 1873 at Vienna the fact was discovered that a dynamo machine of the Gramme type could also act as an electric motor and jwas set in rotation when a current was passed into it from another similar machine.

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  • For a long time Faraday's observation on the rotation of the plane of polarized light by heavy glass in a 1 H.

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  • Kundt prepared metallic films of iron, nickel and cobalt, and obtained powerful negative optical rotation with them (Wied.

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  • Lord Kelvin showed that Faraday's discovery demonstrated that some form of rotation was taking place along lines of magnetic force when passing through a medium.'

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  • Many observers have given attention to the exact determination of Verdet's constant of rotation for standard substances, e.g.

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  • The players, in rotation, must follow suit if able.

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  • By means of a very brisk rotation of cathode, combined with a rapid current of electrolyte, J.

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  • According to the mathematical theory of the instrument,' if V and V' are the potentials of the quadrants and v is the potential of the needle, then the torque acting upon the needle to cause rotation is given by the expression, C(V - V'){v-2(V-{-V')}, where C is some constant.

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  • A system of rotation (cereal, roots, grass) is commonly followed, each division of land lying fallow one year as a rule; not more than two ripe grain-crops are commonly taken consecutively.

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  • This discovery, which gained him the Copley medal of the Royal Society in 1825, was followed by another, that a rotating plate of copper tends to communicate its motion to a magnetic needle suspended over it ("magnetism of rotation").

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  • should not be used for seed, and rotation of crops should be observed as a means of preventing the infection of the crop with the germ.

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  • Rotation must be observed and no diseased sets planted.

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  • Philopoemen transferred the seat of assembly from town to town by rotation, and placed dependent communities on an equal footing with their former suzerains.

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  • Hence leguminous plants are essential in all rotation of crops.

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  • This caused various lateral and contrary movements, resulting in a whirling movement (Slvn) resembling the rotation of Anaxagoras, whereby similar atoms were brought together (as in the winnowing of grain) and united to form larger bodies and worlds.

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  • a, b, Crests of the wave; c, d, e, up strokes; x, x, down strokes; f, point corresponding to the anterior margin of the wing, and forming a centre for the downward rotation of the wing (a, g); g, point corresponding to the posterior margin of the wing, and forming a centre for the upward rotation of the wing (d, f).

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  • " From this description it will be evident that by the mere rotation of the bones of the forearm and hand the maximum and minimum of resistance is FIG.

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  • The compound rotation goes on throughout the entire down and up strokes, and is intimately associated with the power which the wing enjoys of alternately seizing and evading the air.

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  • The compound rotation of the wing is greatly facilitated by the wing being elastic and flexible.

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  • The compound rotation of the wing, as seen in the bird, is represented in fig.

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  • The rotation of the rowing feathers on The rapidity of travel of the insect wing is in some cases enormous.

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  • The flexibility and elasticity of the kite formed by the natural wing are rendered necessary by the fact that the wing, as already stated, is practically hinged at its root and along its anterior margin, an arrangement which necessitates its several parts travelling at different degrees of speed, in proportion as they are removed from the axes of rotation.

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  • v, w, Sockets in which the roots of the blades of the screw rotate, the degree of rotation being limited by steel springs (z, s).

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  • They are carried across the disk by the sun's rotation, partaking in the equatorial acceleration; they also show marked displacements of their own, whether with, or relative to, the neighbouring photosphere does not appear; at the beginning of their life they usually outrun the average daily rotation appropriate to their latitude.

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  • Pending more conclusive evidence from the spectroscope, the interpretation of the peculiar surface rotation of the sun appears to be that the central parts of the body are rotating faster than those outside them; for if such were the case the observed phenomenon would arise.

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  • Its determination is difficult, because it is involved with questions of the changing orientation of the earth's axis of rotation.

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  • Rotation period; latitude o° :24.46d 300:26.43d 60°:29.634 80°:30.56d Solar constant, or units of energy received per minute per square centimetre at earth's mean distance: 2 I calories.

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  • As the uppermost compartment fills, the weights end of the beam rises, and by means of a vertical rod suc - cessively operates on detents connected with the rotation of the two brushes, and stops them in turn.

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  • The ratchet wheel and the disk are automatically connected by clutch mechanism in order to effect the rotation of the brushes.

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  • By the first part of this drop the movement of the poise is suddenly stopped, as will be explained below, and the travel of the poise along the steelyard, which measures the load on the platform, is recorded by the amount of rotation of the large spur wheel, and this is suitably shown on a dial in connexion with the wheel.

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  • If the centre of gravity of a pulley is on the axis of rotation, and the whole mass is distributed so that the axis of inertia coincides with the axis of rotation, there can be no unbalanced force or unbalanced couple as the pulley revolves.

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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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  • Motion in either direction is thereby obtained, and a considerable variation in the speed of rotation can be obtained by providing a cone pulley on the countershaft, which drives the cone pulley secured to the lathe E FIG.

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  • The farms are commonly cultivated on the three-crop rotation system.

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  • The stationary tangent: the line may in the course of its rotation come to rest, and then reverse the direction of its rotation.

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  • That, on account of the earth's rotation, the main part of the Baltic Stream must keep close to the coast of the Scandinavian peninsula.

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  • They used to attend the temple in rotation, and be present at the sacrifices; and as this duty fell to each in his turn, the men of the class or family which he represented were expected in their several cities and places of abode to engage themselves in religious exercises, and especially in fasting.

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  • Among these may be mentioned improved methods of ploughing, tile drainage, use of the press drill, which results in greater immunity against winter killing, crop rotation, and, to a very small extent, fertilization.

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  • Wheat is grown year after year without rotation - except in a few cases - on a third or more of our wheat acreage; not one acre in fifty is directly fertilized for the crop, and only a minimum amount of attention is given to the betterment of seed stock.

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  • Several years before this appointment he had made himself a name by an elegant solution of the problem to find the sun's equator and determine the period of its rotation by observation of the spots on its surface.

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  • He was elected by the Roman people at the close of a year's interregnum, during which the sovereignty had been exercised by the members of the senate in rotation.

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  • He never wanders outside of a certain district, in which he possesses hereditary rights, and maintains a series of camping-grounds which he visits in regular rotation.

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  • (2) When the centre of the earth is taken as origin, the most natural fundamental axis is that of the earth's rotation.

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  • As he travels South, his zenith moves along the celestial sphere, and the circles of diurnal rotation become oblique to the horizon.

    0
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  • In the other class of problems, the relative motion of the different parts of the separate bodies is considered; for example, the rotation of the earth on its axis, and the consequences of the fact that those parts of a body which are nearer to another body are more strongly attracted by it.

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  • This plane remains invariable so long as no third body acts; when it does act the position of the plane changes very slowly, continually rotating round the radius vector of the planet as an instantaneous axis of rotation.

    0
    0
  • But, in consequence of the centrifugal force generated by the rotation, it assumes a spheroidal form, the equatorial regions bulging out.

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    0
  • The point E, being more distant than C, will be attracted with less force, while Q will be attracted with a greater force than will the centre C. Were the force equal on every point of the earth it would have no influence on its rotation, but would simply draw its whole mass toward the attracting body.

    0
    0
  • It is therefore only the difference of the forces on different parts of the earth that affects the rotation.

    0
    0
  • When the orbit of the satellite is inclined to that of the primary planet round the sun, the action brings about a change in the plane of the orbit represented by a rotation round an axis perpendicular to the plane of the orbit of the primary.

    0
    0
  • This is not the direction of gravity proper, or of the earth's attraction, but the resultant of this attraction combined with the centrifugal force due to the earth's rotation on its axis.

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  • Misled, however, into identifying it with magnetism, he imagined circulation in the solar system to be maintained through the material compulsion of fibrous emanations from the sun, carried round by his axial rotation.

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    0
  • C. Vogel's observations on the 9th of June 1871, of differences due to the sun's rotation in the refrangibility of Fraunhofer lines derived respectively from the east and west limbs.

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    0
  • Relatively to the direction of the earth there is really no rotation.

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    0
  • The rate of actual rotation is substantially uniform, while the arc through which the moon moves from day to day varies.

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    0
  • There is also a libration in latitude, arising from the fact that the axis of rotation of the moon is not precisely perpendicular to the plane of her orbit.

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  • 3'46 Ratio of gravity to gravity at the earth's surface 1: 6 Inclination of axis of rotation to ecliptic. ..

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  • In 1856 William Ferrel showed that the action of the moon on the ocean tidal waves would result in a retardation of the earth's rotation, a result, at first unnoticed, which was independently reached a few years later by Delaunay.

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  • The most plausible explanation is that, like the discrepancy in the secular acceleration, the observed deviation is only apparent, and arises from slow fluctuations in the earth's rotation, and therefore in our measure of time produced by the motion of great masses of polar ice and the variability of the amount of snowfall on the great continents.

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  • Each of these ten acted as king for five days, and if, at the end of fifty days, no king had been elected, the rotation was renewed.

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  • Henceforth Athens monopolized one of the two Ionian votes, while the other passed in rotation among the remaining Ionic, perhaps only among the Euboeic, cities.

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  • In the same way Doris held one Dorian vote and the other passed in rotation among the Dorian cities of Peloponnesus; and the east and west Locrians came to have one each.

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  • Arago in 1811 found that in the case of white light and with moderately thin plates the transmitted light is no longer white but coloured, a variation of brightness but not of tint being produced when the polarizer and analyser being crossed are rotated together, while the rotation of the analyser alone produces a change of colour, which passes through white into the complementary tint.

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  • Biot further ascertained that this rotation of the plane of polarization varies as the distance traversed in the plate and very nearly as the inverse square of the wave-length, and found that with certain specimens of quartz the rotation is in a clockwise or right-handed direction to an observer receiving the light, while in others it is in the opposite direction, and that equal plates of the rightand lefthand varieties neutralize one another's effects.

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  • 757) have shown by a few biaxal crystals, such as sugar and Rochelle salt, the rotation produced by a given thickness being in general different, and in some cases of opposite sign for the two optic axes.

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