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axes

axes

axes Sentence Examples

  • Axes and choppers were plied all around.

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  • The steady chinking of ice axes could be heard echoing up and down the deep gorge.

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  • She'd thought his wall of swords, daggers, axes, and other medieval weapons were for ceremony.

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  • A crystal may be regarded as built up of primitive parallelepipeda, the edges of which are in the ratio of the crystallographic axes, and the angles the axial angles of the crystals.

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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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  • (?) and viii., and the satellite of Neptune.); while, to make the argument complete, the planets, so far as they can be observed, rotate on their axes in the same manner.

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  • Orthogonal System.-In particular, if we consider the transformation from one pair of rectangular axes to another pair of rectangular axes we obtain an orthogonal system which we will now briefly inquire into.

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  • One part of it dispersed and waded knee-deep through the snow into a birch forest to the right of the village, and immediately the sound of axes and swords, the crashing of branches, and merry voices could be heard from there.

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  • The cases of greatest practical importance are those of a sphere (which is an ellipsoid with three equal axes) and an ovoid or prolate ellipsoid of revolution.

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  • The hyperbola which has for its transverse and conjugate axes the transverse and conjugate axes of another hyperbola is said to be the conjugate hyperbola.

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  • A wire or rod in this condition is said to be circularly magnetized; it may be regarded as consisting of an indefinite number of elementary ring-magnets, having their axes coincident with the axis of the wire and their planes at right angles to it.

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  • Shrines of the Double Axes have been found in the palace of Cnossus itself, at Hagia Triada, and in a small palace at Gournia, and many specimens of the sacred emblem occurred in the Cave Sanctuary of Dicte, the mythical birthplace of the Cretan Zeus.

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  • The most important subjects of his inquiries are enumerated by Forbes under the following five heads: - (1) The laws of polarization by reflection and refraction, and other quantitative laws of phenomena; (2) The discovery of the polarizing structure induced by heat and pressure; (3) The discovery of crystals with two axes of double refraction, and many of the laws of their phenomena, including the connexion of optical structure and crystalline forms; (4) The laws of metallic reflection; (5) Experiments on the absorption of light.

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  • When crystals are present they generally have their long axes parallel to the fluxion.

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  • When crystals are present they generally have their long axes parallel to the fluxion.

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  • The oldest rocks, the gneisses and schists of the Archean period, form nearly the whole of the Central Plateau, and are also exposed in the axes of the folds in Brittany.

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  • We can distinguish (I) digestive endoderm, in the stomach, often with special glandular elements; (2) circu-, latory endoderm, in the radial and ring canals; (3) supporting endoderm in the axes of the tentacles and in the endodermlamella; the latter is primitively a double layer of cells, produced by concrescence OC-- = w.?"

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  • The main assimilating tissue, on the other hand, is under the upper epidermis, where it is well illuminated, and consists of oblong cells densely packed with chloroplasts and with their long axes perpendicular to the surface (palisade tissue).

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  • Near this domestic quarter was found a small shrine of the Double Axes, with cult objects and offertory vessels in their places.

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  • Great quantities of votive figures and objects of cult, such as the fetish double axes and stone tables of offering, were found both above and below.

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  • If a current is passed through the fixed coil and movable coil in series with one another, the movable coil tends to displace itself so as to bring the axes of the coils, which are normally at right angles, more into the same direction.

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

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  • The cabmen he met and their passengers, the carpenters cutting the timber for new houses with axes, the women hawkers, and the shopkeepers, all looked at him with cheerful beaming eyes that seemed to say: Ah, there he is!

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

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  • The geometry of the rectangular hyperbola is simplified by the fact that its principal axes are equal.

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  • Referred to the asymptotes as axes the general equation becomes xy 2 obviously the axes are oblique in the general hyperbola and rectangular in the rectangular hyperbola.

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  • Complete scenes of worship in which libations are poured before the Sacred Axes are, moreover, given on a fine painted sarcophagus found at Hagia Triada.

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  • The chest is of limestone coated with stucco, adorned with life-like paintings of offertory scenes in connexion with the sacred Double Axes of Minoan cult.

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  • In the theory of surfaces we transform from one set of three rectangular axes to another by the substitutions 'X=' by+ cz, Y = a'x + b'y + c'z, Z =a"x+b"y-l-c"z, where X 2+Y2+Z2 = x2+ y2+z2.

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  • If the magnetization is parallel to the major axis, and the lengths of the major and minor axes are 2a and 2C, the poles are situated at a distance equal to 3a from the centre, and the magnet will behave externally like a simple solenoid of length 3a.

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  • The collection was made up of Shipton's newly purchased, barely used, ice climbing gear, ropes, ice axes, pitons and various garments.

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  • The distance between the centres of the two spectrographs shall be equal to the distance between the optical axes of the two viewing microscopes.

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  • The whole district of Casas Grandes is further studded with artificial mounds, from which are excavated from time to time large numbers of stone axes, metates or corn-grinders, and earthern vessels of various kinds.

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  • This gives access to a whole series of halls and private rooms (halls " of the Colonnades," " of the Double Axes," " Queen's Megaron" with bath-room attached and remains of the fish fresco, " Treasury " with ivory figures and other objects of art), together with extensive remains of an upper storey.

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  • The equivalent volumes and topic parameters are tabulated: From these figures it is obvious that the first three compounds form a morphotropic series; the equivalent volumes exhibit a regular progression; the values of x and t,t, corresponding to the a axes, are regularly increased, while the value of w, corresponding to the c axis, remains practically unchanged.

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  • 4 S'N' is a small magnet of moment M', and SN a distant fixed magnet of moment M; the axes of SN and S'N' make angles of 0 and 4 respectively with the line through their middle points.

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  • The distance between the centres of the two spectrographs shall be equal to the distance between the optical axes of the two viewing microscopes.

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  • 4 S'N' is a small magnet of moment M', and SN a distant fixed magnet of moment M; the axes of SN and S'N' make angles of 0 and 4 respectively with the line through their middle points.

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  • This quarter of the palace shows the double axe sign constantly repeated on its walls and pillars, and remains of miniature wall-paintings showing pillar shrines, in some cases with double axes stuck into the wooden columns.

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  • This quarter of the palace shows the double axe sign constantly repeated on its walls and pillars, and remains of miniature wall-paintings showing pillar shrines, in some cases with double axes stuck into the wooden columns.

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  • The platy minerals have also a perfect cleavage parallel to their flat surfaces, while the fibrous species often have two or more cleavages following their long axes; hence a schistose rock may split not only by separation of the mineral plates from one another but also by cleavage of the parallel minerals through their substance.

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  • To reduce these figures to a common standard, so that the volumes shall contain equal numbers of molecules, the notion of molecular volumes is introduced, the arbitrary values of the crystallographic axes (a, b, c) being replaced by the topic parameters' (x, ?i, w), which are such that, combined with the axial angles, they enclose volumes which contain equal numbers of molecules.

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  • Ammonium iodide assumes cubic forms with perfect cubic cleavage; tetramethyl ammonium iodide is tetragonal with perfect cleavages parallel to {100} and {o01} - a difference due to the lengthening of the a axes; tetraethyl ammonium iodide also assumes tetragonal forms, but does not exhibit the cleavage of the tetramethyl compound; while tetrapropyl ammonium iodide crystallizes in rhombic form.

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  • The angle between the optic axes varies from 70 0 -50° in muscovite and lepidolite to Io - o° in biotite and phlogopite; the latter are thus frequently practically uniaxial.

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  • ax., Green axis creeping on the surface of damp soil; rh., colorless rhizoids penetrating the soil; asc. ax., ascending axes of green cells.

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  • 6), be the axes of temperature and pressure, and A corresponds to the transition point (95.6°) of rhombic sulphur, we may follow out the line AB which shows the elevation of the transition point with increasing pressure.

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  • Or, instead of looking upon a linear substitution as replacing a pencil of lines by a projectively corresponding pencil retaining the same axes of co-ordinates, we may look upon the substitution as changing the axes of co-ordinates retaining the same pencil.

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  • two by the choice of axes, two by the choice of multiples.

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  • 3) by F, and its components parallel to the co-ordinate axes by X and Y, we have X= - ax = M(3 cos' 0 - I), Y= - y = M (3 sin 0 cos 0.

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

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  • According to this theory the molecules of any magnetizable substance are little permanent magnets the axes of which are, under ordinary conditions, disposed in all possible directions indifferently.

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  • If V denote the potential, F the resultant force, X, Y, Z, its components parallel to the co-ordinate axes and n the line along which the force is directed, then - sn = F, b?= X, - Sy = Y, -s Surfaces for which the potential is constant are called equipotential surfaces.

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  • According to this theory the molecules of any magnetizable substance are little permanent magnets the axes of which are, under ordinary conditions, disposed in all possible directions indifferently.

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  • If the leaf of Mini osa or Desmodium be examined, it will be seen that at the base of each leaflet and each leaf, just at the junction with the respective axes, is a swelling known as a pulvinus.

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  • of supporting axes from assimilating appendages, and as the body increases in size and becomes a solid mass of cells or interwoven threads, a corresponding differentiation of a superficial assimilative system from the deep-lying parts.

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  • A special form of this " baetylic " cult in Minoan Crete was the representation of the two principal divinities in their fetish form by double axes.

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  • The sternum has no keel, and ossifies from lateral and paired centres only; the axes of the scapula and cora.coid have the same general direction; certain of the cranial bones have characters very unlike those possessed by the next order - the vomer, for example, being broad posteriorly and generally intervening between the basisphenoidal rostrum and the palatals and pterygoids; the barbs of the feathers are disconnected; there is no syrinx or inferior larynx; and the diaphragm is better developed than in other birds.'

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  • The internal force F is opposite to the direction of the magnetization, and equal to NI, where N is a coefficient depending only on the ratio of the axes.

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  • There are, however, always three principal axes at right angles to one another along which the magnetization and the force have the same direction.

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  • If each of these axes successively is placed parallel to the lines of force in a uniform field H, we shall have = 12 = 13=K3H, the three susceptibilities being in general unequal, though in some cases two of them may have the same value.

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  • principal axes of the crystal, the actual magnetization will be the resultant of the three magnetizations along the axes.

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  • On the other hand, the form of the third curve, with its large intercepts on the axes of H and B, denotes that the specimen to which it relates possesses both retentiveness and coercive force in a high degree; such a metal would be chosen for making good permanent magnets.

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  • Weber's theory, the molecules of a ferromagnetic metal are small permanent magnets, the axes of which under ordinary conditions are turned indifferently in every direction, so that no magnetic polarity is exhibited by the metal as a whole; a magnetic force acting upon the metal tends to turn the axes of the little magnets in one direction, and thus the entire piece acquires the properties of a magnet.

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  • The strength of the induced current is - HScosO/L, where 0 is the inclination of the axis of the circuit to the direction of the field, and L the coefficient of self-induction; the resolved part of the magnetic moment in the direction of the field is equal to - HS 2 cos 2 6/L, and if there are n molecules in a unit of volume, their axes being distributed indifferently in all directions, the magnetization of the substance will be-3nHS 2 /L, and its susceptibility - 3S 2 /L (Maxwell, Electricity and Magnetism, § 838).

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  • When, as in the application to rectangular or circular apertures, the form is symmetrical with respect to the axes both of x and y, S = o, and C reduces to C = ff cos px cos gy dx dy,.

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  • Let x, y, z be the co-ordinates of any particle of the medium in its natural state, and, 7 7, the displacements of the same particle at the end of time t, measured in the directions of the three axes respectively.

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  • (b2V2 + n2) (a2 - b 2) = - z It will now be convenient to introduce the quantities a l, a 2', 7731 which express the rotations of the elements of the medium round axes parallel to those of co-ordinates, in accordance with the equations Ty - 1 = dz ' 3= - dy 2 = dx - In terms of these we obtain from (7), by differentiation and subtraction, (b 2 v 2 + n 2) 7,3 = 0 (b 2 0 2 +n 2) .r i = dZ/dy (b 2 v 2 +n 2)', , 2 = - dZ/dx The first of equations (9) gives 3 = 0 (10) For al we have ?1= 47rb2, f dy e Y tkr dx dy dz

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  • Their weapons consisted of bow and arrows, short swords, spears and axes.

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  • Among the industries of Belfast are trade with the surrounding country, the manufacture of shoes, leather boards, axes, and sashes, doors and blinds, and the building and repairing of boats.

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  • Successive flexures or ridges are ranged in more or less parallel lines, and from between the bands of hard, unyielding rock of older formation the soft beds of recent shale have been washed out, to he carried through the enclosing ridges by rifts which break across their axes.

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  • The coefficient of expansion is constant for such metals only as crystallize in the regular system; the others expand differently in the directions of the different axes.

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  • But by Green's transformation f flpdS = f f PPdxdydz, (2) thus leading to the differential relation at every point = dy dp The three equations of equilibrium obtained by taking moments round the axes are then found to be satisfied identically.

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  • (9) Turning the axes to make them coincide with the principal axes of the area A, thus making f f xydA = o, xh = - a 2 cos a, y h = - b 2 sin a, (io) where ffx2dA=Aa2, ffy 2 dA= Ab 2, (II) a and b denoting the semi-axes of the momental ellipse of the area.

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  • In particular, with sh a =1, the cross-section of n = a is x 4 +6x 2 y 2 4= 2c 4, or x 4 -{-y =c 4 (20) when the axes are turned through 45°.

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  • Moving Axes in Hydrodynamics.

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  • The components of velocity of the moving origin are denoted by U, V, W, and the components of angular velocity of the frame of reference by P, Q, R; and then if u, v, w denote the components of fluid velocity in space, and u', v', w' the components relative to the axes at a point (x, y, z) fixed to the frame of reference, we have u =U +u' - yR +zQ, v =V +v -zP +xR, w=W +w -xQ +yP.

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  • .) = (X-jfidp) +m (Y-dy) +n (Zp 2), for all values of 1, m, n, leading to the equations of motion with moving axes.

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  • As an application of moving axes, consider the motion of liquid filling the ellipsoidal case 2 y 2 z2 Ti + b1 +- 2 = I; (1) and first suppose the liquid be frozen, and the ellipsoid l3 (4) (I) (6) (9) (I o) (II) (12) (14) = 2 U ¢ 2, (15) rotating about the centre with components of angular velocity, 7 7, f'; then u= - y i +z'i, v = w = -x7 7 +y (2) Now suppose the liquid to be melted, and additional components of angular velocity S21, 522, S23 communicated to the ellipsoidal case; the additional velocity communicated to the liquid will be due to a velocity-function 2224_ - S2 b c 6 a 5 x b2xy, as may be verified by considering one term at a time.

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  • If u', v', w' denote the components of the velocity of relative to the axes, = u +yR - zQ =a2+ b2S23y - c2 a2 ?

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  • The hydrodynamical equations with moving axes, taking into account the mutual gravitation of the liquid, become dp +4 p Ax+ du - vR {-wQ?

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  • (17) ellipsoid of liquid of three unequal axes, rotating bodily about the least axis;.

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  • Beginning with a single body in liquid extending to infinity, and denoting by U, V, W, P, Q, R the components of linear and angular velocity with respect to axes fixed in the body, the velocity function takes the form = Ucb1+V42+W43+ P xi+Qx2+Rx3, (I) where the 0's and x's are functions of x, y, z depending on the shape of the body; interpreted dynamically, C -p0 represents the impulsive pressure required to stop the motion, or C +p4) to start it again from rest.

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  • These equations are proved by taking a line fixed in space, whose direction cosines are 1, then dt=mR-nQ,' d'-t = nP =lQ-mP. (5) If P denotes the resultant linear impulse or momentum in this direction P =lxl+mx2+nx3, ' dP dt xl+, d y t x2' x3 +1 dtl dt 2 +n dt3, =1 ('+m (dt2-x3P+x1R) ' +n ('-x1Q-{-x2P) ' '= IX +mY+nZ, / (7) for all values of 1, Next, taking a fixed origin and axes parallel to Ox, Oy, Oz through 0, and denoting by x, y, z the coordinates of 0, and by G the component angular momentum about 1"2 in the direction (1, G =1(yi-x2z+x3y) m 2-+xlz) n(y(y 3x 1 x3x y + x 2 x) (8) Differentiating with respect to t, and afterwards moving the fixed.

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  • the precedin investigation, the liquid stops dead when the body is brought to rest and when the body is in motion the surrounding liquid moves in uniform manner with respect to axes fixed in the body, and the force experienced by the body from the pressure of the liquid on it surface is the opposite of that required to change the motion of the liquid; this has been expressed by the dynamical equations give above.

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  • Its axes measured 505 and 404 ft.

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  • An ample supply of natural gas is utilized by its manufacturing establishments; and among its manufactures are axes, lumber, foundry and machine shop products, furniture, boilers, woollen goods, glass and chemical fire-engines.

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  • Bituminous coal, natural gas and oil abound in the vicinity; the river provides excellent water-power; the borough is a manufacturing centre of considerable importance, its products including iron and steel bridges, boilers, steam drills, carriages, saws, files, axes, shovels, wire netting, stoves, glass-ware, scales, chemicals, pottery, cork, decorative tile, bricks and typewriters.

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  • 2 Consider then an ellipsoidal shell the axes of whose bounding surfaces are (a, b, c) and (a+da), (b+db), (c+dc), where da/a =db b =dc/c =,u.

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  • The above expressions for the capacity of an ellipsoid of three unequal axes are in general elliptic integrals, but they can be evaluated for the reduced cases when the ellipsoid is one of revolution, and hence in the limit either takes the form of a long rod or of a circular disk.

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  • The isothermals are approximately equilateral hyperbolas (pv= constant), with the axes of p and v for asymptotes, for a gas or unsaturated vapour, but coincide with the isopiestics for a saturated vapour in presence of its liquid.

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  • I by the whole area B"DZ'VO under the isothermal 9"D and the adiabatic DZ', bounded by the axes of pressure and volume.

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  • Sca, through,, u rpov, measure), in geometry, a line passing through the centre of a circle or conic section and terminated by the curve; the "principal diameters of the ellipse and hyperbola coincide with the "axes" and are at right angles; " conjugate diameters " are such that each bisects chords parallel to the other.

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  • In the cliffs opposite the town on the south is the rock-cut church of the Madonna del Parto, developed, no doubt, out of an Etruscan tomb, of which there are many here; and close by is a rock-hewn amphitheatre of the Roman period, with axes of 55 and 44 yds., now most picturesque.

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  • The slides are kept firmly down to their bearings by the rollers r, r, r, r, attached to axes which are, in the middle, very strong springs.

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  • The former contrivances consist essentially of levers or cams with toothed surfaces or gripping shoes mounted upon transverse axes attached to the sides of the cage, whose function is to take hold of the guides and support the cage in the event of its becoming detached from the rope.

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  • The opposite axes are connected with springs which are kept in compression by tension of the rope in drawing but come into action when the pull is released, the side axes then biting into wooden guides or gripping those of steel bars or ropes.

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  • &prtos, even, and &Lktvxos, a finger or toe, "even-toed"), the suborder of ungulate mammals in which the central (and in some cases the only) pair of toes in each foot are arranged symmetrically on each side of a vertical line running through the axes of the limbs.

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  • '}'anent) vessel, and let co-ordinate axes be taken such that the origin is in dS, and the axis of x is the normal at the origin into the gas.

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  • Now if the atoms are regarded as points or spherical bodies oscillating about positions of equilibrium, the value of n+3 is precisely six, for we can express the energy of the atom in the form (9 2 v a 2 v a2v E = z(mu 2 +mv 2 +mw 2 +x 2 ax2 + y2ay2-fz2az2), where V is the potential and x, y, z are the displacements of the atom referred to a certain set of orthogonal axes.

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  • The pattern is that of a true sight, that is to say, the base plate is capable of movement about two axes, one parallel to and the other at right angles to the axis of the gun, and has cross spirit-levels and a graduated elevating drum and independent deflection scale, so that compensation for level of wheels can be given and quadrant elevation.

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  • Hence the area of an ellipse whose axes are 2a and 2b is Trab; and the volume of an ellipsoid whose axes are 2a, 2b and 2c is t rabc. The area of a strip of an ellipse between two lines parallel to an axis, or the volume of the portion (frustum) of an ellipsoid between two planes parallel to a principal section, may be found in the same way.

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  • In the case of the briquette the position of the foot of the ordinate u is expressed by co-ordinates x, y, referred to a pair of axes parallel to a pair of sides of the base of the briquette.

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  • This value of 0 is the same for all parabolas which pass through D and E and have their axes at right angles to KL.

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

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  • These coils are placed with their axes at right angles to one another, and at the point where the axes intersect a small pivoted needle of soft iron is placed, carrying a longer index needle moving over a scale.

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  • a point which divides a line, or a line which divides an angle, into two equal parts; in crystallography it denotes the bisector of the angle between the optic axes.

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  • Their axes were made of tough stones, sawn from the block and ground to the fitting shape.

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  • They bought practically all of what is now Essex county from the Indians for "fifty double hands of powder, one hundred bars of lead, twenty axes, twenty coats, ten guns, twenty pistols, ten kettles, ten swords, four blankets, four barrels of beer, ten pairs of breeches, fifty knives, twenty horses, eighteen hundred and fifty fathoms of wampum, six ankers of liquor (or something equivalent), and three troopers' coats."

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  • Now the expression above given cannot be integrable exactly, under all circumstances and whatever be the axes of co-ordinates, unless (�2u',�2vi,�2w') is the gradient of a continuous function.

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  • In the simplest case, that of uniform translation, these components of the gradient will each be constant throughout the region; at a distant place in free aether where there is no motion, they must thus be equal to -u,-v,-w, as they refer to axes moving with the matter.

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  • We may now, as is somewhat the more natural course in the terrestrial application, take axes (x,y,z) which move with the matter; but the current must be invariably defined by the flux across surfaces fixed in space, so that we may say that relation (i) refers to a circuit fixed in space, while (ii) refers to one moving with the matter.

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  • For the simplest case of polarized waves travelling parallel to the axis of x, with the magnetic oscillation y along z and the electric oscillation Q along y, all the quantities are functions of x and t alone; the total current is along y and given with respect to our moving axes by __ (d_ d Q+vy d K-1 Q, dt dx) 47rc 2 + dt (4?rc 2) ' also the circuital relations here reduce to _ dydQ _dy _ dx 47rv ' _ dt ' d 2 Q dv dx 2 -417t giving, on substitution for v, d 2 Q d 2 Q d2Q (c2-v2)(7372 = K dt 2 2u dxdt ' For a simple wave-train, Q varies as sin m(x-Vt), leading on substitution to the velocity of propagation V relative to the moving material, by means of the equation KV 2 + 2 uV = c 2 v2; this gives, to the first order of v/c, V = c/K i - v/K, which is in accordance with Fresnel's law.

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  • In south Oran they determine the principal axes of the mountain ranges.

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  • Stone axes, remains of carved stone pillars similar to those of Easter Island, and skeletons with a pearl-mussel beneath the head have been found in the island, though it was uninhabited when discovered by Philip Carteret in 1767.

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  • When the currents flow through the two coils, forces are brought into action compelling the coils to set their axes in the same direction, and these forces can be opposed by another torque due to the control of a spiral spring regulated by moving a torsion head on the instrument.

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  • Copper was known to them, and it is possible that they knew how to make cutting instruments from it, but they generally used stone axes, hammers and picks, and their most dangerous weapon was a war-club into which chips of volcanic glass were set.

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  • marvellous series of rock folds with gently undulating axes, trending north-east and south-west through a belt 70 or 80 m.

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  • high, half a mile or a mile long, with axes parallel to the direction of the ice motion as indicated by striae on the underlying rock floor; these hills are known by the Irish name, drumlins, used for similar hills in north-western Ireland.

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  • The mountains rapidly grow wider and higher northward, by taking on new complications of structure and by including large basins between the axes of uplift, tintil in northern Colorado and Utah a complex of ranges has a breadth of 300 m., and in Colorado alone there are 40 summits over 14,000 ft.

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  • two for each pair of circles, and it may be shown that these lie three by three on four lines, named the " axes of similitude."

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  • The generality of treatment is indeed remarkable; he gives as the fundamental property of all the conics the equivalent of the Cartesian equation referred to oblique axes (consisting of a diameter and the tangent at its extremity) obtained by cutting an oblique circular cone in any manner, and the axes appear only as a particular case after he has shown that the property of the conic can be expressed in the same form with reference to any new diameter and the tangent at its extremity.

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  • While it is throughout essentially a mountainous country, very complicated in its orographic features and interlocking river systems, two principal mountain axes form its ruling features - the Rocky Mountains proper, above referred to, and the Coast Ranges.

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  • e, f, Anterior portions of these axes fused by concrescence to the wall of the body.

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  • 15, where g and h are respectively the left and the right ctenidial axes projecting freely beyond the body.

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  • axes; but, in the same way also the results are correct if the resolution is treated as an analytical device and in the final result account is taken of all the overlapping components.

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  • Proposition 14 shows how to draw an ellipse through five given points, and Prop. 15 gives a simple construction for the axes of an ellipse when a pair of conjugate diameters are given.

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  • This arrangement is not very convenient, as it is difficult to protect the mirror from accidental displacement, so that the angle between the geometrical and magnetic axes may vary.

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  • The axis of the magnet is horizontal and at the same level as the mirror magnet, while when the central division of the scale B appears to coincide with the vertical cross-wire of the telescope the axes of the two magnets are at right angles.

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  • What is known as the method of sines is used, for since the axes of the two magnets are always at right angles when the mirror magnet is in its zero position, the ratio M/H is proportional to the sine of the angle between the magnetic axis of the mirror magnet and the magnetic - = meridian.

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  • Hence, when the microscopes are adjusted so as to coincide with the points of the dipping needle a, the axes of the two needles must be at right angles.

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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 strike of these folds is usually east and west and roughly parallel to the axes of elevation of the plateau.

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  • Every river which rises in Tibet washes down sands impregnated with gold, and it has been proved that this gold is not the product of intervening strata, but must have existed primarily in the crystalline rocks of the main axes of upheaval.

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  • It is usual to take three axes at right angles to each other to represent pressure, temperature and the composition of the variable phase..

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  • The "tangential arcs" (T) were explained by Young as being caused by the thin plates with their axes horizontal, refraction taking place through alternate faces.

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  • The axes will take up any position, and consequently give rise to a continuous series of parhelia which touch externally the inner halo, both above and below, and under certain conditions (such as the requisite altitude of the sun) form two closed elliptical curves; generally, however, only the upper and lower portions are seen.

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  • The axis P N is made of varnished glass, and so are the axes that join the three plates with the brass axis N 0.

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  • THEODOLITE,' a surveying instrument consisting of two graduated circles placed at right angles to each other, for the measurement of horizontal and vertical angles, a telescope, which turns on axes mounted centrically to the circles, and an alidade for each circle, which carries two or more verniers.

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  • Crystals exhibit pyroelectrical characters, since they possess four uniterminal triad axes of symmetry.

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  • Other objects found in the graves are small flint knives, stone axes, flint and lumps of pyrites for obtaining fire, and, in the womens graves, hand-mills for grinding corn.

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  • Flint axes were made in imitation of metal in the XIIth Dynasty (9).

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  • Fighting Weapons.The battle-axe has been described above with axes.

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  • AXES.

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  • On extending his inquiry to other aelotropic crystals he observed a similar variation, and was thus led, in 1825, to the discovery that aelotropic crystals, when heated, expand unequally in the direction of dissimilar axes.

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  • In the following year he discovered the change, produced by change of temperature, in the direction of the optic axes of selenite.

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  • The city lies in an agricultural and grape-growing; region, and has a fine harbour and an extensive lake trade; the: manufactures include locomotives, radiators, lumber, springs, shirts, axes, wagons, steel, silk gloves and concrete blocks.

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  • They have been thrown into many folds, the long axes of which run in a general north-easterly direction.

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  • The Danes were specially renowned for their axes; but about the sword the most of northern poetry and mythology clings.

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  • The simplest equation to the parabola is that which is referred to its axis and the tangent at the vertex as the axes of co-ordinates, when it assumes the form y 2 = 4ax where as = semilatus rectum; this may be deduced directly from the definition.

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  • An equation of similar form is obtained when the axes of co-ordinates are any diameter and the tangent at the vertex.

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  • In crystallography, the regular or ordinary dodecahedron is an impossible form since the faces cut the axes in irrational ratios; the "pentagonal dodecahedron" of crystallographers has irregular pentagons for faces, while the geometrical solid, on the other hand, has regular ones.

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  • that Sur la theorie des variations des elements des planetes, et en particulier des variations des grands axes de leurs orbites.

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  • The peculiar form of the tube is eminently suited for rigid preservation of the relative parallelism of the axes of the two telescopes, so that,;i the image of a certain selected star is retained on the intersection of two wires of the micrometer, by means of the driving clock, aided by small corrections given by the observer in right ascension and declination (required on account of irregularity in the clock movement, error in astronomical adjustment of the polar axis, or changes in the star's apparent place produced by refraction), the image of a star will continue on the same spot of the photographic film during the whole time of exposure.

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  • The driving circle was greatly increased in diameter and placed at the upper end of the polar axis, and both the polar and declination axes were made much stronger in proportion to the mass of the instrument they were designed to carry.

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  • - T he modern equatorial should, for general purposes, be capable of carrying spectroscopes of considerable weight, so that the proportional strength of the axes and the rigidity of the instrument have to be considerably increased.

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  • focal length (described in Washington Observations, 1874, App. 1) was in these respects very defective, the polar and declina tion axes being only 7 in.

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  • arranged, and are all necessary for the quick and easy working raised and lowered nearly in an arc of a circle of which the point of intersection of the polar and declination axes is the centre.

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  • They then meet a small plane mirror supported at the point of intersection of the polar and declination axes, whence they are reflected down through the hollow polar axis as shown in fig.

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  • When the angular momentum is too great for the usual spheroidal form to persist, this gives place to an ellipsoid with three unequal axes; this is succeeded by a pear-shaped form.

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  • The graves at Hallstatt were partly inhumation partly cremation; they contained swords, daggers, spears, javelins, axes, helmets, bosses and plates of shields and hauberks, brooches, various forms of jewelry, amber and glass beads, many of the objects being decorated with animals and geometrical designs.

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  • The weapons and axes are mostly iron, a few being bronze.

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  • The flat axes are distinguished by the side stops and in some cases the transition from palstave to socketed axe can be seen.

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  • Quaternions (as a mathematical method) is an extension, or improvement, of Cartesian geometry, in which the artifices of co-ordinate axes, &c., are got rid of, all directions in space being treated on precisely the same terms. It is therefore, except in some of its degraded forms, possessed of the perfect isotropy of Euclidian space.

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  • Servois says, with reference to the general representation of a directed line in space: " L'analogie semblerait exiger que le trineme filt de la forme p cos a+q cos a+r cos y; a, 1 3, y etant les angles d'une droite avec trois axes rectangulaires; et qu'on eut (p cos a+ g cos /3+ r cos y)(p' cos a+ q cos /3 + r cos y) =cos 2 a+cos 2) 3+cos 2 y = 1.

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  • Hamilton, still keeping prominently before him as his great object the invention of a method applicable to space of three dimensions, proceeded to study the properties of triplets of the form x+iy+jz, by which he proposed to represent the directed line in space whose projections on the co-ordinate axes are x, y, z.

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  • Also, everything relating to change of systems of axes, as for instance in the kinematics of a rigid system, where we have constantly to consider one set of rotations with regard to axes fixed in space, and another set with regard to axes fixed in the system, is a matter of troublesome complexity by the usual methods.

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  • Here the symmetry points at once to the selection of the three principal axes as the directions for i, j, k; and it would appear at first sight as if quaternions could not simplify, though they might improve in elegance, the solution of questions of this kind.

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  • it is independent of the particular directions chosen for the rectangular co-ordinate axes.

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  • the line parallel to q' q-- 1 which intersects the axes of Q and Q'; the plane of the member contains a fixed line; the centre is on a fixed ellipse which intersects the transversal; the axis is on a fixed ruled surface to which the plane of the ellipse is a tangent plane, the ellipse being the section of the ruled surface by the plane; the ruled surface is a cylindroid deformed by a simple shear parallel to the transversal.

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  • In the third-order complex the centre locus becomes a finite closed quartic surface, with three (one always real) intersecting nodal axes, every plane section of which is a trinodal quartic. The chief defect of the geometrical properties of these bi-quaternions is that the ordinary algebraic scalar finds no place among them, and in consequence Q:1 is meaningless.

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  • Mercury and Venus were also studied, and he concluded that these planets rotated on their axes in the same time as they revolved about the sun; but these views are questioned.

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  • It is hardly necessary to say that the latter procedure has hitherto been found to be adequate~ As a first step we adopt a system of rectangular axes whos origin is fixed in the earth, but whose directions are fixed b)

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  • At each step there is a gain in itccu racy and comprehensiveness; and the conviction is cherishei that some system of rectangular axes exists with respeci to which the Newtonian scheme holds with all imaginabb accuracy.

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  • Moving axes of reference.

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  • Adopting rectangular axes Ox, Oy, in the plane of, f~ the forces, arid distinguishing FIG the various forces of the system 4.

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  • by suffixes, we can replace the system by two forces X, Y, in the direction of co-ordinate axes; viz.

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  • acting along the co-ordinate axes.

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  • If the axes are rectangular, the direction-ratios become direction-cosines, so that X1 + ~s2 + vi = I, whence R2 = X1 + ~2 + Z2.

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  • If the origin of rectangular axes fixed in the lamina be shifted through a space whose projections on the original directions of the axes are X, u, and if the axes are simultaneously turned through an angle e, the coordinates of a point of the lamina, relative to the original axes, are changed from x, y to X+x cos ey sin e, u+x sin e+y cos e, or X + x ye, u + Xe + y, ultimately.

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  • The force at A1 may be replaced by its components Xi, Yi, parallei to the co ordinate axes; that at A1 by V1 its components X2, Yf, and so on.

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

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  • Successive half-turns about parallel axes a, b are equivalent to a translation measured by double the distance between these axes in the direction from a to 1,.

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

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  • It is easily inferred as a limiting case, or proved directly, that two infini tesimal rotations a, j3 about c u parallel axes are equivalent to a ..._ -- - -

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  • so that a+f~=o, the point C is at infinity, and the effect is a translation perpendicular to tire plane of the two given axes, of amount a.

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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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  • Siace E2 + if + ~1, or ef, is necessarily an absolute invariant for all transformations of the (rectangular) co-ordinate axes, we infer that XE + un + v~ is also an absolute invariant.

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  • We will briefly notice the case of two degrees, which involves an interesting generalization of the method (already explained) of compounding rotations about intersecting axes.

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  • We examine first the case where the axes of the two screws are at right angles and intarsect.

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  • We take these as axes of x and y; then if f, n be the component rotations about them, we have -

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  • In the first place, a cyhindroid can be constructed so as to have its axis coincident with the common perpendicular to the axes of the two given screws and to satisfy thi-ee other conditions, for the position of the centre, the parameter, and the orientation about the axis are still at our disposal.

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  • Hence we can adjust these so that the surface shall contain the axes of the two given screws as generators, and that the difference of the corresponding pitches shall have the proper value.

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  • It follows that when a body has two degrees Of freedom it can twist about any one of a singly infinite system of screws whose axes lie on a certain cylindroid.

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  • Hence a couple of moment G, whose axis has the direction (1, m, n) relative to a right-handed system of rectangular axes, FIG.

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  • It is seen that L~, Mi, Ni are the moments of the original force at P~ about the co-ordinate axes.

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  • In words: the sum of the projections of the forces on each of the co-ordinate axes must vanish; and, the sum of the moments of the forces about each of these axes must vanish.

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  • Let the axes of the 1~~1T

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  • If we take rectangular axes Ox, Oy, of which Oy is drawn vertically upwards, we have y=sin ~ s, whence T=wy.

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  • If the axes of x and y be taken horizontal and vertical (upwards), we derive x =a log (sec #+tan ~), y= a sec ~.

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  • relative to any axes, rectangular or oblique be (x1,1 yi, Zi), (xi, yi, Zf),..

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  • If we take rectangular axes through any point 0, the quadratic moments with respect to the co-ordinate planes are I,, = Z(mxi), I,,= Z(my1), I, = ~(mz2); (9) those with respect to the co-ordinate axes are Ii,, = ~lm(y~+z2)~, I,, = ~tm(z2+x2)l, I,, ~tm(x2+y1)j; (10) whilst the polar quadratic moment with respect to 0 is 10 = ~tm(x2+y2+z1)}.

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  • The method of homogeneous strain can be applied to deduce the corresponding results for an ellipsoid of semi-axes a, b, c. If the co-ordinate axes coincide with the principal axes, we find l0=1/2Ma2, I9=~Mb2, I~ = ~ Me2, whence Ii.~ =3/4M (b1 +ci), &c.

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  • The formula (16) expresses that the squared radius of gyration about any axis (Ox) exceeds the squared radius of gyration about a parallel axis through G by the square of the distance between the two axes.

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  • If the co-ordinate axes coincide with the principal axes of this quadric, we shall have ~(myz) =0, ~(mzx) =0, Z(mxy) = 0~ (24) and if we write ~(mx) = Ma, ~(my1) = Mb, ~(mz) =Mc2, (25) where M=~(m), the quadratic moment becomes M(aiX2+bI,s2+ cv), or Mp, where p is the distance of the origin from that tangent plane of the ellipsoid ~-,+~1+~,=I, (26)

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  • It appears from (24) that through any assigned point 0 three rectangular axes can be drawn such that the product of inertia with respect to each pair of co-ordinate planes vanishes; these are called the principal axes of inertia at 0.

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  • It may further be shown that if Binets ellipsoid be referred to any system of conjugate diameters as co-ordinate axes, its equation will be ~2+~2+~-2I, (27)

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  • Since the quadratic moments with respect to w and of are equal, it follows that w is a plane 01 stationary quadratic moment at P, and therefore a principal plane of inertia at P. In other words, the principal axes 01 inertia at P are the normals to the three confocals of the systen (3,~) which pass through P. Moreover if x, y, z be the co-ordinates of P, (33) is an equation to find the corresponding values of 0; and if Of, 02, 03 be the roots we find Oi+O2+81r1a2$-7, (35)

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  • The squares of the radii of gyration about the principal axes at P may be denoted by k,i+k32, k,f + ki2, k12 + k,2 hence by (32) and (35), they are rfOi, r2Oi, r20s, respectively.

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  • To find the relations between the moments of inertia about different axes through any assigned point 0, we take 0 as origin.

    0
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  • A, B, C are the moments of inertia about the co-ordinate axes, and F, G, H are the products of inertia with respect to the pairs of co-ordinate planes.

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  • When referred to its principal axes, the equation of the quadric takes the form Axi+By2+Czi=M.

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

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  • If we write A=Ma, B=M/32, C=M~y, the formula (37), when referred to the principal axes at 0, becomes if p denotes the perpendicular drawn from 0 in the direction (X, u, e) to a tangent plane of the ellipsoid ~+~+~=I (43)

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  • Obviously OV is parallel to the tangent to the path atP, and its magnitude is ds/dt, where s is the arc. If we project OV on the co-ordinate axes (rectangular or oblique) in the usual manner, the projections u, v, w are called the component velocities parallel to the axes.

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  • If X, Y, Z are the components of force, then considering the changes in an infinitely short time 3t we have, by projection on the co-ordinate axes, i3(mu) =Xi5t, and so on, or du dv dw m-~jj=X, m~=Y, m~=Z.

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    0
  • If u be the acceleration at unit distance, the component accelerations parallel to axes of x and y through 0 as origin will be ux, uy, whence ~ = ~sy.

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    0
  • The path is therefore an ellipse of which a, b are conjugate semi-diameters, and is described in the period 24 Ju; moreover, the velocity at any point P is equal to ~ OD, where OD is the semi-diameter conjugate to OP. ~,This type of motion;,s called elliptic harmonic. If the co-ordinate axes are the principal axes of the ellipse, the angle ft in (I o) is identical with the excentric angle.

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  • The above problem is identical with that of the oscillation of a particle in a smooth spherical bowl, in the neighborhood of the lowest point, If the bowl has any other shape, the axes Ox, Oy may, ..--7 be taken tangential to the lines tof curvature ~ / at the lowest point 0; the equations of small A motion then are dix xdiy (II) c where P1, P2, are the principal radii of curvature at 0.

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    0
  • If the axes of x and y be drawn horizontal and vertical (upwards), and if ~ be the inclination of the tangent to the horizontal, we have dv.

    0
    0
  • Thus taking nny point 0 as base, we have first a linear momentum whose components referred to rectangular axes through 0 are ~(m~), Z(m~), ~(mb); - (I)

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    0
  • Secondly, we have an angular momentum whose components are ~{m(y~z3)}, ~lm(z~xb)1, ~{m(xi?yi~)}, (2) these being the sums of the moments of the momenta of the several particles about the respective axes.

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  • (6) If we put ~, ~Y, s=o, the theorem is proved as regards axes parallel to Ox.

    0
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  • For different parallel axes, the period of a small oscillation varies as ~i, or ~f (GO+OP); this is least, subject to the condition (4), when GO=GP=,c. The reciprocal relation between the centres of suspension and oscillation is the basis of Katers method of determining g experimentally.

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

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  • The theory of principal or permanent axes was first investigated from this point of view by J.

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  • by 7 (5); hence, if ~, u, v be now used to denote the component angular momenta about the co-ordinate axes, we have X=~tm(pyqx)ym(rxpz)zl, with two similar formulae, or x= ApHqGr=~, 1

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    0
  • ,j If the co-ordinate axes be taken to coincide with the principal axes of inertia at 0, at the instant under consideration, we have the simpler formulae 2T=Api+Bqi+Cri, (8)

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    0
  • The relation between these axes may be expressed by means of the momental ellipsoid at 0.

    0
    0
  • The equation of the latter, referred to its principal axes, being as in II (41), the co-ordinates of the point J where it is met by the instantaneous axis are proportional to p, q, r, and the direction-cosines of the normal at J are therefore proportional to Ap, Bq, Cr, or X, u, v.

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  • where, for simplicity, the co-ordinate axes are supposed to coincide with the principal axes at the mass-centre.

    0
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  • L, M, N) denotes the system of extraneous forces referred (like the momenta) to the mass-centre as base, the co-ordinate axes being of course fixed in direction.

    0
    0
  • The resulting Z+R equations are not as a rule easy of application, owing to the fact that the moments and products of inertia A, B, C, F, G, H are not constants but vary in conse- 0 quence of the changing orientation of the body with respect to the co-ordinate axes.

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

    0
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  • Let a be the radius of the rolling sphere, c that of the spherical surface which is the locus of its centre, and let x, y, I be the co-ordinates of this centre relative to axes through 0, the centre of the fixed sphere.

    0
    0
  • If p, q, r be the component angular velocities about the principal axes at 0, we have (Ap+B2q+C,2)/r = (Ap+Bq1+Cr2)/2T, (3) each side being in fact equal to unity.

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

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    0
  • Moving A xes of ReferenceFor the more general treatment of the kinetics of a rigid body it is usually convenient to adopt a system of moving axes.

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  • In order that the moments and products of inertia with respect to these axes may be constant, it is in general necessary to suppose them fixed in the solid.

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  • The moving axes Ox, Oy, 01 form a rigid frame of reference whose motion at time t may be specified by the three component angular velocities p, q, r.

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    0
  • Now consider a system of fixed axes Ox, Oy, Oz chosen so as to coincide at the instant I with the moving system Ox, Oy, Os.

    0
    0
  • If we now apply them to the case of a rigid body moving about a fixed point 0, and make Ox, Oy, Oz coincide with the principal axes of inertia at 0, we have X, u, v=Ap, Bq, Cr, whence A (B C) qr = L,

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

    0
    0
  • When the origin of the moving axes is also in motion with a velocity whose components are u, v, w, the dynamical equations are ~rn+qi=X, ~pl+rE=Y, ~f_qE+Pi,=Z, (II)

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    0
  • j To prove these, we may take fixed axes Ox, Oy, Oz coincident with the moving axes at time t, and compare the linear and angular momenta E+E, ~ ~ ?~+~X, u+u, v+~v relative to the new position of the axes, Ox, Oy, Oz at time t+t with the original momenta ~, ~ ~, A, j~i, v relative to Ox, Oy, Oz at time t.

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    0
  • When, in any problem, the values of u, v, w, p, q, r have been determined as functions of t, it still remains to connect the moving axes with some fixed frame of reference.

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  • If OA, OB, OC be principal axes of inertia of a solid, and if A, B, C denote the corresponding moments of inertia, the kinetic energy is given by 2TA(~ sin 4,sin 0 cos 44~)2+B Ce cos 4,+sin0 sin$)i +C (~+cos0~)2.

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  • This solution, taken by itself, represents a motion in which each particle of the system (since its displacements parallel to Cartesian co-ordinate axes are linear functions of the qs) executes a simple vibration of period 21r/u.

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