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radius

radius

radius Sentence Examples

  • The radius of curves for the 1 .

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  • Atlanta was laid out in the form of a circle, the radius being 14 m.

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  • The distance of the lucid points was the tangent of the magnified angles subtended by the stars to a radius of io ft.

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  • The distance of the lucid points was the tangent of the magnified angles subtended by the stars to a radius of io ft.

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  • As the entire time required for light to pass over the radius of the earth's orbit is only about 500 seconds, this error is fatal to the method.

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  • As the entire time required for light to pass over the radius of the earth's orbit is only about 500 seconds, this error is fatal to the method.

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  • radius and gradients of 1 in 50.

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  • A semicircle is then drawn behind the bowls with a radius of 9 ft.

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  • Curves of still smaller radius have entailed a maximum speed of io m.

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  • For a radius of 30 m.

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  • The chief difference between the first three types lies in the weight of rails and rolling stock and in the radius of the curves.

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  • Houses within a radius of 3 m.

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  • - An electric current i flowing uniformly through a cylindrical wire whose radius is a produces inside the wire a magnetic field of which the lines of force are concentric circles around the axis of the wire.

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  • From the law of angular motion of the latter its radius vector will run ahead of PQ near A, PQ will overtake and pass it at apocentre, and the two will again coincide at pericentre when the revolution is completed.

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  • an hour and a minimum radius of curve of 300 ft.

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  • As there was not a single town or large village in the vicinity of the camp, the immense number of generals and courtiers accompanying the army were living in the best houses of the villages on both sides of the river, over a radius of six miles.

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  • This metropolitan area, within a radius of approximately io m.

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  • radius lies on the lesser heights between Langstrath and Dunmail Raise, which may, however, be the crown of an ancient dome of rocks, "the dissected skeleton of which, worn by the warfare of air and rain and ice, now alone remains" (Dr H.

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  • If a hollow sphere 7 of which the outer radius is R and the inner radius r is placed in a uniform field Ho, the field inside will also be uniform and in the same direction as Ho, and its value will be approximately 3 i - R 3 For a cylinder placed with its axis at right angles to the lines of force, 2 = Ho (41) 2 +4(-2)(i - R2) These expressions show that the thicker the screen and the greater its permeability o, the more effectual will be the shielding action.

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  • On the lines actually authorized by the Board of Trade under the 1896 act the normal minimum radius of the curves has been fixed at about 600 ft.; when a still smaller radius has been necessary, the speed has been reduced to 10 m.

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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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  • each radius, in all eight separate gonad-masses, as the name implies.

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  • each radius, in all eight separate gonad-masses, as the name implies.

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  • The radius of a 1-degree curve is 5730 ft., or about 861 chains, of a 15-degree curve 383 ft.

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  • The circles A'A', which are struck with 2-inch radius, define the first portion of the knuckle.

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  • From the former we deduce Ho, and from the latter the corresponding value of I, using the formulae Ho = 47rin/l and I - X s, (d (-- 11)2n7rr 2 i where s is the deflection in scale-divisions, n the distance in scaledivisions between the scale and the mirror, and r the radius of the wire.

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  • From the former we deduce Ho, and from the latter the corresponding value of I, using the formulae Ho = 47rin/l and I - X s, (d (-- 11)2n7rr 2 i where s is the deflection in scale-divisions, n the distance in scaledivisions between the scale and the mirror, and r the radius of the wire.

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  • For, if x be its radius, we have {(r+2X)2 - x2} +.v {a 2_, 21= so that x 2 = Xar/(a+r) nearly.

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  • Where the face of the warehouse is sufficiently close to the water to permit of the crane rope plumbing the hatches without requiring a jib of excessive radius, it is a very convenient plan to place the whole crane on the warehouse roof.

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  • radius, while on over 100 m.

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  • According to the assumed law of the secondary wave, the result must actually depend upon the precise radius of the outer boundary of the region of integration, supposed to be exactly circular.

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  • Pop. (1901), 9137; within the five-mile radius, 14,087.

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  • It is shown in the paper that the greatest possible force which the isthmus method can apply at a point in the axis of the bobbin is F = 11, 137 I, log i n b/a, I, being the saturation value of the magnet pores, a the radius of the neck on which the cones converge, and b the radius of the bases of the cones.

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  • When therefore sensible uniformity is desired, the radius of the ring should he large in relation to that of the convolutions, or the ring should have the form of a short cylinder with thin walls.

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  • It is served by the Pennsylvania, the Baltimore & Ohio, and the Wheeling & Lake Erie railways, and is connected by an interurban electric system with all the important cities and towns within a radius of 50 m.

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

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  • (It is easy to see that the radius of the bright spot is of the same order of magnitude.) The experiment succeeds in a dark room of the length above mentioned, with a threepenny bit (supported by three threads) as obstacle, the origin of light being a small needle hole in a plate of tin, through which the sun's rays shine horizontally after reflection from an external mirror.

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

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  • This is a very good and comparatively cheap form of crane, where a long and variable radius is required, but it cannot slew through a complete circle.

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  • The wing of the bird is folded in a unique way, namely, the radius parallel with the humerus, and the whole wrist and hand with their ulnar side against the ulna; upper and forearm in a state of supination, the hand in that of strong abduction.

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  • In Great Britain the curvature is defined by stating the length of the radius, expressed in chains (i chain=66 ft.), in America by stating the angle subtended by a chord ioo ft.

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  • On some of the earlierEnglish main lines no curves were constructed of a less radius than a mile (80 chains), except at places where the speed was likely to be low, but in later practice the radius is sometimes reduced to 40 or 30 chains, even on high-speed passenger lines.

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  • The amount of superelevation required to prevent derailment at a curve can be calculated under perfect running conditions, given the radius of curvature, the weight of the vehicle, the height of the centre of gravity, the distance between the rails, and the speed; but great experience 1 See The Times Engineering Supplement (August 22, 1906), p. 265.

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  • The inner circle B has a radius of 12 in.

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  • 2.6 in.) radius; width of formation, 3.50 m.

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  • The anomaly is then the angle BFP which the radius vector makes with the major axis.

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  • The metropolitan system, which extends around the city on a radius of 10 to 12 m., was begun in 1893.

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  • The problem of finding a radius vector satisfying this condition is one which can be solved only by successive approximations, or tentatively.

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  • radius, but outside the city limits proper, there was a further population equal in number to that within the municipality itself.

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  • m., a circle with radius of 15 m.

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  • If the coil has the form of a ring of mean radius r, the length will be 21rr, and the field inside the coil may be expressed as H = 2ni/r.

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  • The potential due to a uniformly magnetized sphere of radius a for an external point at a distance r from the centre is V =:I ra 3 I cos 0/r 22, (23) 0 being the inclination of r to the magnetic axis.

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  • This is a very good and comparatively cheap form of crane, where a long and variable radius is required, but it cannot slew through a complete circle.

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  • If the coil has the form of a ring of mean radius r, the length will be 21rr, and the field inside the coil may be expressed as H = 2ni/r.

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

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  • radius it was 20,976.

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  • U, ulna; R, radius; c, cuneiform; 1, lunar; s, scaphoid; u, unciform; m, magnum; td, trapezoid; tm, trapezium.

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  • Learning, however, that these were still beyond striking radius, he determined to deal with Mack's army first, having formed the fixed conviction that a threat at the latter's communications would compel him to endeavour to retreat southwards towards Tirol.

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  • By Kepler's second law the radius vector, FP, sweeps over equal areas in equal times.

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  • We will now investigate the total illumination distributed over the area of the circle of radius r.

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  • In any case the proportion of the whole illumination to be found outside the circle of radius r is given by J02(z)+J12(z).

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  • Verdet has compared Foucault's results with theory, and has drawn the conclusion that the radius of the visible part of the image of a luminous point was equal to half the radius of the first dark ring.

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  • The derrick crane introduces a problem for which many solutions have been sought, that of preventing the load from being lifted or lowered when the jib is pivoted up or down to alter the radius.

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  • The distal end of the humerus ends in a trochlea, with a larger knob for the ulna and a smaller oval knob for the radius.

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  • r, Radius.

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  • The curves on railways are either simple, when they consist of a portion of the circumference of a single circle, or compound, when they are made up of portions of the circumference of two or more circles of different radius.

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  • But if the change from straight to circular is made through the medium of a suitable curve it is possible to relieve the abruptness, even on curves of comparatively small radius.

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  • In the great continental basin there are long lines with easy gradients and curves, while in the Allegheny and Rocky Mountains the gradients are stiff, and the curves numerous and of short radius.

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  • Let P, P' be two consecutive positions of the radius vector.

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  • If, however, the primary wave be spherical, and of radius a at the wave-front of resolution, then we kno* that at a distance r further on the amplitude of the primary wave will be diminished in the ratio a:(r+a).

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  • It is thus sufficient to determine the intensity along the axis of p. Putting q = o, we get C = ffcos pxdxdy=2f+Rcos 'px 1/ (R2 - x2)dx, R being the radius of the aperture.

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  • The value of C for an annular aperture of radius r and width dr is thus dC =271-Jo(Pp)pdp, (12).

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  • Again, if we compare the complete circle with a narrow annular aperture of the same radius, we see that in the latter case the first dark ring occurs at a much smaller obliquity, viz.

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  • The same method of representation is applicable to spherical waves, issuing from a point, if the radius of curvature be large; for, although there is variation of phase along the length of the infinitesimal strip, the whole effect depends practically upon that of the central parts where the phase is sensibly constant.'

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  • 17 APQ is the arc of the circle representative of the wavefront of resolution, the centre being at 0, and the radius OA being equal to a.

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  • Generally only one bow is clearly seen; this is known as the primary rainbow; it has an angular radius of about 410, and exhibits a fine display of the colours of the spectrum, being red on the outside and violet on the inside.

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  • Its angular radius is about J7°.

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  • The mathematical discussion of Airy showed that the primary rainbow is not situated directly on the line of minimum deviation, but at a slightly greater value; this means that the true angular radius of the bow is a little less than that derived from the geometrical theory.

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  • In the same way, he showed that the secondary bow has a greater radius than that previously assigned to it.

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  • 656s, a way, and yp&4*t y, to write), a curve of which the radius vector is proportional to the velocity of a moving particle.

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  • In the endless-rope systems cars run singly or in short trains, curves are disadvantageous, unless of long radius, speed is relatively slow, and branch roads not so easily operated as with tail-rope.

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  • On starting to hoist, the rope winds from the small towards the large end of the drum, the lever arm, or radius of the coils, increasing as the weight of Winding Engine.

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  • The cartesian equation referred to the axis and directrix is y=c cosh (x/c) or y = Zc(e x / c +e x / c); other forms are s = c sinh (x/c) and y 2 =c 2 -1-s 2, being the arc measured from the vertex; the intrinsic equation is s = c tan The radius of curvature and normal are each equal to c sec t '.

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  • The blower then heats the end of the cylinder again and rapidly spins the pipe about its axis; the centrifugal effect is sufficient to spread the soft glass at the end to a radius equal to that of the rest of the cylinder.

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  • As the molten metal is run in, the upward thrust on the outside mould, when the level has reached PP', is the weight of metal in the volume generated by the revolution of APQ; and this, by a theorem of Archimedes, has the same volume as the cone ORR', or rya, where y is the depth of metal, the horizontal sections being equal so long as y is less than the radius of the outside FIG.

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  • In the more general case of the convective equilibrium of a spherical atmosphere surrounding the earth, of radius a, (1-1?-=(n+ I) Po --a 2 dr, (12) gravity varying inversely as the square of the distance r from the centre; so that, k = po/po, denoting the height of the homogeneous atmosphere at the surface, 0 is given by (n+I)k(I -9/6 0) =a(I -a/r), (13) or if c denotes the distance where 0=o, 0 _a (14) 0 r c -a' When the compressibility of water is taken into account in a deep ocean, an experimental law must be employed, such as p - po=k(P - Po), or P/po=I+(p-p0)/A, A=kpo, (15) so that A is the pressure due to a head k of the liquid at density under atmospheric pressure po; and it is the gauge pressure required on this law to double the density.

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  • The varying direction of the inclining couple Pc may be realized by swinging the weight P from a crane on the ship, in a circle of radius c. But if the weight P was lowered on the ship from a crane on shore, the vessel would sink bodily a distance P/wA if P was deposited over F; but deposited anywhere else, say over Q on the water-line area, the ship would turn about a line the antipolar of Q with respect to the confocal ellipse, parallel to FF', at a distance FK from F FK= (k2-hV/A)/FQ sin QFF' (2) through an angle 0 or a slope of one in m, given by P sin B= m wA FK - W'Ak 2V hV FQ sin QFF', (3) where k denotes the radius of gyration about FF' of the water-line area.

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  • if r denotes the radius of curvature of the stream line, so that I dp + dV - dH _ dq 2 q2 (6) p dv dv dv dv - r ' the normal acceleration.

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  • For instance, in a uniplanar flow, radially inward towards 0, the flow across any circle of radius r being the same and denoted by 27rm, the velocity must be mfr, and 0=m log r,, y=m0, +4,i =m log re ie, w=m log z.

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  • Over a concentric cylinder, external or internal, of radius r=b, 4,'=4,+ Uly =[U I - + Ui]y, (4) and 4" is zero if U 1 /U = (a 2 - b2)/b 2; (5) so that the cylinder may swim for an instant in the liquid without distortion, with this velocity Ui; and w in (I) will give the liquid motion in the interspace between the fixed cylinder r =a and the concentric cylinder r=b, moving with velocity U1.

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  • Along the path of a particle, defined by the of (3), _ c) sine 2e, - x 2 + y2 = y a 2 ' (Io) sin B' de' _ 2y-c dy 2 ds ds' on the radius of curvature is 4a 2 /(ylc), which shows that the curve is an Elastica or Lintearia.

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  • (7) Thus with g=o, the cylinder will describe a circle with angular velocity 2pw/(a+p), so that the radius is (a+p)v/2pw, if the velocity is v.

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  • The velocity of a liquid particle is thus (a 2 - b 2)/(a 2 +b 2) of what it would be if the liquid was frozen and rotating bodily with the ellipse; and so the effective angular inertia of the liquid is (a 2 -b 2) 2 /(a 2 +b 2) 2 of the solid; and the effective radius of gyration, solid and liquid, is given by k 2 = 4 (a 2 2), and 4 (a 2 For the liquid in the interspace between a and n, m ch 2(0-a) sin 2E 4) 1 4Rc 2 sh 2n sin 2E (a2_ b2)I(a2+ b2) = I/th 2 (na)th 2n; (8) and the effective k 2 of the liquid is reduced to 4c 2 /th 2 (n-a)sh 2n, (9) which becomes 4c 2 /sh 2n = s (a 2 - b 2)/ab, when a =00, and the liquid surrounds the ellipse n to infinity.

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  • For instance, with n = I in equation (9), the relative stream function is obtained for a sphere of radius a, by making it, y' =1y+2Uy 2 = 2U(r 2 -a 3 /r) sin?

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  • (to) Integrating over the base, to obtain one-third of the kinetic energy T, 3T = 2 pf '3 4R2(3x4-h4)dx/h 3 = pR2h4 / 1 35 V 3 (II) so that the effective k 2 of the liquid filling the trianglc is given by k 2 = T/Z p R 2 A = 2h2/45 = (radius of the inscribed circle) 2, (12) or two-fifths of the k 2 for the solid triangle.

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  • 5, I ch S2 = u a, sh C2= ' u (I) (I and along the jet APJ, oo > u=aerslc>a, sh S2=i sin 0 =iu=ie zrs/o, (2) PM sin 0 ds = f e ds = 1 = 1 sin 0, (3) cos 272a - cos 2n0 = 2Q - ?ib L a b2 s i n' 27ta u-b A (a- (u -a.u -b') sin 2110 - 2 a-a .u-b  ?l (u -a.u -a') = s in 2na u-b 2n b) A (ab.ba') p l u -bJ (u -a.u -a') sh nS2=i sin 110=i then the radius sin 2170 (30) A', cos nO= i, sin n0=o, n 1 ' b-a' ch nS2= ch log (9) = Va -a' n shnS2= shlog (Q) q _ o> u>a'.

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  • The image of a source of strength p at S outside a sphere of radius a is a source of strength pa/f at H, where 'OS' =f, OH =a2/f, and a line sink reaching from the image H to the centre 0 of line strength - A la; this combination will be found to produce no flow across the surface of the sphere.

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  • When S and S' lie on the same radius, taken along Ox, the Stokes' function can; be written down; and when S and S' coalesce a doublet is produced, with a doublet image at H.

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  • Thus for m =2, the spheres are orthogonal, and it can be verified that a13 a2 3 aY3 i f /' = ZU (I - 13 - 7.2 3 + 3) ' (8) where a l, a2, a =a l a 2 /J (a 1 2 +a 2 2) is the radius of the spheres and their circle of intersection, and r 1, r 2, r the distances of a point from their centres.

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  • the moment of inertia of the body about the axis, denoted by But if is the moment of inertia of the body about a mean diameter, and w the angular velocity about it generated by an impluse couple M, and M' is the couple required to set the surrounding medium in motion, supposed of effective radius of gyration k', If the shot is spinning about its axis with angular velocity p, and is precessing steadily at a rate about a line parallel to the resultant momentum F at an angle 0, the velocity of the vector of angular momentum, as in the case of a top, is C i pµ sin 0- C2µ 2 sin 0 cos 0; (4) and equating this to the impressed couple (multiplied by g), that is, to gN = (c 1 -c 2)c2u 2 tan 0, (5) and dividing out sin 0, which equated to zero would imply perfect centring, we obtain C21 2 cos 0- (c 2 -c 1)c2u 2 sec 0 =o.

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  • Sometimes the cells are erected in a circle, so that the spout below the slicing machine revolving above them with a corresponding radius can discharge the slices into the centre of any of the cells.

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  • Radii are drawn from the centre of the quadrant to the points of division of the arc, and these radii are intersected by the lines drawn parallel to BC and through the corresponding points on the radius AB.

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  • The quadratrix of Tschirnhausen is constructed by dividing the arc and radius of a quadrant in the same number of equal parts as before.

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  • having a charge Q repels a unit charge placed at a distance x from its centre with a force Q/x 2 dynes, and therefore the work W in ergs expended in bringing the unit up to that point from an infinite distance is given by the integral W = Q x 2 dx = Hence the potential at the surface of the sphere, and therefore the potential of the sphere, is Q/R, where R is the radius of the sphere in centimetres.

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  • capacity of the sphere in free space is Q/V = R, or is numerically the same as its radius reckoned in centimetres.

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  • Let R 1 be the radius of the inner sphere, R2 the inside radius of the outer sphere, and R2 the outside radius of the outer spherical shell.

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  • This last result shows that the capacity of a thin disk is 2/7r =1/1.571 of that of a sphere of the same radius.

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  • pp. 137 and 347) determined in 1773 experimentally that the capacity of a sphere was 1.541 times that of a disk of the same radius, a truly remarkable result for that date.

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  • p. 112) has given a full expression for the capacity C of two circular plates of thickness t and radius r placed at any distance d apart in air from which the edge effect can be calculated.

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  • Let a solid circular sectioned cylinder of radius R 1 be enclosed in a coaxial tube of inner radius R2.

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  • radius.'

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  • The electric force due to a point-charge q at a distance r is defined to be q/r 2, and the total flux or induction through the sphere of radius r is therefore 41rq.

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  • 6) of radius r, and centre at C, and let P be any point on it.

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  • A census taken in July 1896 showed a population within a radius of 3 m.

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  • He gave no objective, and when the brigadier pointed out that the enemy was still beyond the striking radius of his horses, Frossard reiterated the order, which was obeyed to the letter.

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  • In the fore-limb the upper and lower series of carpal bones scarcely alternate, but in the hindfootthe astragalus overlaps the cuboid, while the fibula, which is quite distinct from the tibia(as is the radius from the ulna in the fore-limb), articulates with both astragalus and calcaneum.

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  • (X 8.) have existed since the Early U, ulna; R, radius; c, cuneiform; Eocene period.

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  • At a radius of nearly a mile is another wall within which lies the closely-packed city proper, and beyond which the town stretches away to the royal parks on the north and to the business quarter, the warehouses, rice-mills, harbour and docks on the south.

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  • in radius, and with New Castle as the centre.

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  • 5, p. iv.) he found that the index error of the scale changed systematically in different position angles by quantities which were independent of the direction of gravity relative to the position angle under measurement, but which depended solely on the direction of the measured position angle relative to a fixed radius of the object-glass.

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  • On the other hand it is not necessary to reset the telescope after each reversal of the segments.4 When Bessel ordered the Konigsberg heliometer, he was anxious to have the segments made to move in cylindrical slides, of which the radius should be equal to the focal length of the object-glass.

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  • At any point a sounding line would hang in the line of the radius of curvature of the water surface.

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  • This method, which is the oldest, is best adapted for ways that are nearly level, or when many branches are intended to be worked from one engine, and can be carried round curves of small radius without deranging the trains; but as it is intermittent in action, considerable engine-power is required in order to get up the required speed, which is from 8 to ro m.

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  • Reduction of the ulna from a complete and distinct bone to a comparatively rudimentary state in which it coalesces more or less firmly with the radius.

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  • B, Red deer (Cervus elaphus), R, Radius.

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  • The odontoid process of the second vertebra is pig-like: and the tibia and fibula and radius and ulna are severally distinct.

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  • The general theory of this kind of brake is as follows: - Let F be the whole frictional resistance, r the common radius of the rubbing surfaces, W the force which holds the brake from turning and whose line of action is at a perpendicular distance R from the axis of the shaft, N the revolutions of the shaft per minute, co its angular velocity in radians per second; then, assuming that the adjustments are made so that the engine runs steadily at a uniform speed, and that the brake is held still, clear of the stops and without oscillation, by W, the torque T exerted by the engine is equal to the frictional torque Fr acting at the brake surfaces, and this is measured by the statical moment of the weight W about the axis of revolution; that is T =Fr=WR...

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  • Both these forces usually act at the same radius R, the distance from the axis to the centre line of the rope, in which case the torque T is (W-p)R, and consequently the brake horse-power is (W - p)RX21rN, When µ 33,000 changes the weight W rises or falls against the action of the spring balance until a stable condition of running is obtained.

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  • This device consists of a roller of radius r, pressed into contact with a disk.

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  • the relative angular displacement is proportional to the radius of the circle described by the end of a light lever operated by mechanism between the spring-connected parts.

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  • When a shaft is driven by means of gearing the driving torque is measured by the product of the resultant pressure P acting between the wheel teeth and the radius of the pitch circle of the wheel fixed to the shaft.

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  • What is given by the formulae is accordingly the mean radius of an irregularly shaped solid (or, more probably, of the region in which the field of force surrounding such a solid is above a certain intensity), and the mean has to be taken in different ways in the different phenomena.

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  • It extended over a circular area, with a radius of 50 m.

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  • within a radius of 30 m.

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  • The defences consist of an inner line of works which preserve the place against surprise, and of an outlying chain of detached forts of fairly modern construction, forming roughly two-thirds of a circle of three miles radius.

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  • called the sighting radius.

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  • If the dispart sight were EarlyTangent being used, the sighting radius would be OD, but, as Sight.

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  • The formula for length of scale is, length = sighting radius X tangent of the angle of elevation.

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  • This was arranged for by a movable leaf carrying the sighting V, worked by means of a mill-headed screw provided with a scale in degrees and fractions to the same radius as the elevation scale, and an arrowb head for reading.

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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 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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  • The ideas of the centre and of the constancy of the radius do not, however, enter into the elementary conception of the circle as a round figure.

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  • Denoting the constant ratio by fir, the area of a circle is ira 2, where a is the radius, and ir=3.14159 approximately.

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  • The expression 27ra for the length of the circumference can be deduced by considering the limit of the area cut off from a circle of radius a by a concentric circle of radius a - a, when a becomes indefinitely small; this is an elementary case of differentiation.

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  • In the case of the sphere, for instance, whose radius is R, the area of the section at distance x from the centre is lr(R 2 -x 2), which is a quadratic function of x; the values of So, Si, and S2 are respectively o, 7rR 2, and o, and the volume is therefore s.

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  • The centroid of a hemisphere of radius R, for instance, is the same as the centroid of particles of masses 0, 7rR 2, and 4.

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    0
  • Let a be the radius of a circle, and 0 (circular measure) the unknown angle subtended by an arc. Then, if we divide 0 into m equal parts, and L 1 denotes the sum of the corresponding chords, so that L i =2ma sin (0/2m), the true length of the arc is L1 +a9 3 - 5 + ..., where cp. =B/2m.

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  • radius, is drawn on a card as in fig.

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  • From these points as centres, circles are drawn in succession, each with radius greater than the last by a fixed amount, say 4 or 5 mm.

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  • In the figure the radius of the inner circle is 3 mm.

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  • and the radii of the circles drawn round it are 12, 16, 20, &c. If the figure thus drawn is spun round its centre in the right direction in its own plane waves appear to travel out from the centre along any radius.

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

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  • 12 represent a horizontal section of the dome through the source P. Let OPA be the radius through P. Let PQ represent a ray of sound making the angle B with the tangent at A.

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  • Then the reflected ray QR and the ray reflected at R, and so on, will all touch the circle drawn with ON as radius.

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  • Hence all rays between =0 will be confined in the space between the outer dome and a circle of radius OP cos 0, and the weakening of intensity will be chiefly due to vertical spreading.

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

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  • The wave from D has travelled to a circle of radius nearly equal to DF, that from C to a circle of radius nearly CF, and so on.

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  • § 307) that for a cylindrical tube of radius R, provided with a flat extended flange, the loop may be regarded as about o 82 R, in advance of the end.

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  • To obtain the virtual length we must add the correction for each open end, probably about I 2 radius.

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  • Both obtained the value for the velocity (U I C RA(21rNp ' where U is the velocity in free air, R is the radius of the pipe, N the frequency, and p the air density.

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  • Let us suppose that the rod is circular, of radius r, and that the radial displacement of the surface is r t.

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  • The mass of matter moving through A per second is pwa 2 U, where a is the radius of the wire and p is its density.

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  • The radius of gyration of the section is 2a 2.

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  • Substituting in (33) we get U 2 = n/p. (34) If we now keep the wire at rest the disturbance travels along it with velocity U= d (nip), and it depends on the rigidity and density of the wire and not upon its radius.

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  • Every point is equidistant from a fixed point within the surface; this point is the "centre," the constant distance the "radius," and any line through the centre and intersecting the sphere is a "diameter."

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  • Calling the radius r, and denoting by the ratio of the circumference to the diameter of a circle, the volume is 31rr 3, and the surface 41rr2.

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  • If the centre be (a, a, y), the Cartesian equation becomes (x - a) 2 l3)2 + (z - y)2 = a2; consequently the general equation is x2+y2 -}- z 2 + 2Ax+ 2By+2Cz+D =o, and it is readily shown that the co-ordinates of the centre are (-A, -B, -C), and the radius A2+B2+C2-D.

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  • that there exists a point such that the tangents from this point to the four spheres are equal, and that with this point as centre, and the length of the tangent as radius, a sphere may be described which cuts, the four spheres at right angles; this "orthotomic" sphere corresponds to the orthogonal circle of a system of circles.

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  • radius cxxvii.

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  • If w is the weight of a locomotive in tons, r the radius of curvature of the track, v the velocity in feet per sec.; then the horizontal force exerted on the bridge is wv 2 /gr tons.

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  • Then the deflection at the centre is the value of y for x = a, and is _ 5 wa4 S - 14 EI' The radius of curvature of the beam at D is given by the relation R=EI/M.

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  • 72 with arcs of the length 1,, l2, l3, &c., and with the radii r1, r 2, &c. (note, for a length 2l 1 at each end the radius will be infinite, and the curve must end with a straight line tangent to the last arc), then let v be the measured deflection of this curve from the straight line, and V the actual deflection of the bridge; we have V = av/b, approximately.

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  • in length would be drawn equal to to in.; then if the true radius at the centre were to,000 ft., this radius, if the curve were undistorted, 1, FIG.

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  • would be on paper woo in., but making 50 we can draw the curve with a radius of 20 in.

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  • Under Austria, since everywhere that 40 scholars of one nationality were to be found within a radius of 5 km.

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  • By electric lines it is connected with most of the cities and towns within a radius of 20 m., including Jersey City, Paterson and the residential suburbs, among which are the Oranges, Montclair, Bloomfield, Glen Ridge, Belleville and Nutley.

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  • C,C', D,D', two types of medusa organization; C and D are composite sections, showing a radius (R) on one side, an interradius (IR) on the other; C' and D' are plans; the mouth and manubrium are indicated at the centre, leading into the gastral cavity subdivided by the four areas of concrescence in each interradius (IR).

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  • In the territory within a radius of 10 m.

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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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  • If the pressure falls off uniformly, so that the pressure-curve is a straight line PDF sloping downwards and cutting AM in F, then the energy-curve will be a parabola curving downwards, and the velocity-curve can be represented by an ellipse, or circle with centre F and radius FA; while the time-curve will be a sinusoid.

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  • Especially within a radius of 20 m.

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  • The soil within a radius of some 20 m.

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  • The manufacture of cement was begun in 1829 at Shippingport, a suburb of Louisville, whence the natural cement of Kentucky and Indiana, produced within a radius of 15 m.

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  • - Nathaniel Roe's Tabulae logarithmicae (1633) was the first complete seven-figure 1 In describing the contents of the works referred to, the language and notation of the present day have been adopted, so that for example a table to radius 10,000,000 is described as a table to 7 places, and so on.

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  • radius is another circle - to the east Kew (9469) and Hawthorne (21,430), to the south-east St Kilda (20,542) and Brighton (10,047), to the south-west Williamstown (14,052) and Footscray (18,318), to the north-west Essenden (17,426), and Flemington and Kensington (10,946), and to the north Brunswick (24,141).

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  • Among the more prominent of many others that are admired for their beauty are Squam, New Found, Sunapee and Ossipee, all within a radius of a few miles from Winnepesaukee; Massabesic farther S.; and Diamond Ponds, Umbagog and Connecticut lakes, N.

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  • Howth, Malahide and Sutton to the north, and Bray to the south, are, favoured seaside watering-places outside the radius of actual suburbs.

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  • radius and there had been constant patrol skirmishes and small actions which continued until the Armistice.

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  • lined with planking, having a radius of 9 ft.

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  • CG, the " radius."

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  • The co-ordinates of its centre are - g/c, f/c; and its radius is (g 2 +f 2 - c) I.

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  • Since the equation to a circle of zero radius is x 2 +y 2 =o, i.e.

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  • The general equation to a circle in this system of co-ordinates is deduced as follows: If p be the radius and 1p+mg+nr=o the centre, we have p= (lpl+mgi+nri)/(l+m+n), in which i, q i, r i is a line distant p from the point 1p+mq+nr= o.

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  • This equation, which may be more conveniently written tap, bq, cr} 2 = (Ap+pq+vr) 2, obviously represents a circle, the centre being Xp+µqt-vr=o, and radius 20/p(X+µ+v).

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  • A system coaxal with the two given circles is readily constructed by describing circles through the common points on the radical axis and any third point; the minimum circle of the system is obviously that which has the common chord of intersection for diameter, the maximum is the radical axis - considered as a circle of infinite radius.

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  • To construct circles coaxal with the two given circles, draw the tangent, say XR, from X, the point where the radical axis intersects the line of centres, to one of the given circles, and with centre X and radius XR describe a circle.

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  • Very early in the history of geometry it was known that the circumference and area of a circle of radius r could be expressed in the forms 27rr and 7rr2.

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  • Data: radius = a.

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  • Area of sector =100 = z radius X arc.

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  • Since the area of a circle equals that of the rectilineal triangle whose base has the same length as the circumference and whose altitude equals the radius (Archimedes, KIKXou A ir, prop.i), it follows that, if a straight line could be drawn equal in length to the circumference, the required square could be found by an ordinary Euclidean construction; also, it is evident that, conversely, if a square equal in area to the circle could be obtained it would be possible to draw a straight line equal to the circumference.

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  • 7 angled at C, ADB is the semicircle described on AB as diameter, AEB the circular arc described with centre C and radius CA= CB.

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  • 9): ACB being a semicircle whose centre is 0, and AC the arc to be rectified, he produced AB to D, making BD equal to the radius, joined DC, 1 Vieta, Opera math.

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  • If this be applied to the right-hand side of the identity m m m 2 m2 tan-=- - n n -3n-5n" it follows that the tangent of every arc commensurable with the radius is irrational, so that, as a particular case, an arc of 45 having its tangent rational, must be incommensurable with the radius; that is to say, 3r/4 is an incommensurable number."

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  • All the wings are of firm, glassy texture, and very complex in their neuration; a remarkable and unique feature is that a branch of the radius (the radial sector) crosses the median nervure, while, by the development of multitudinous cross-nervules, the wing-area becomes divided into an immense number of small areolets.

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  • The femur has a small third trochanter, the radius and ulna and tibia and fibula are respectively separate, at least in the young, and the fibula articulates with the astragalus.

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  • of the building and one guinea for those outside this radius.

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  • radius from New Castle, Delaware, forms the boundary between it and Delaware.

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  • 4, by an average radius of 1200 Dutch fathoms (7874 ft.) from the outer glacis of the fortress.

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  • the summer is propagated by the mosquito (Anopheles claviger) marks a new epoch; the most diverse theories as to its origin had hitherto been propounded, but it is now possible to combat it on a definite plan, by draining the marshes, protecting the houses by fine mosquito-proof wire netting (for Anopheles is not active by day), improving the water supply, &c., while for those who have fever, quinine (now sold cheaply by the state) is a great specific. A great improvement is already apparent; and a law carried in 1903 for the Bonifica dell' Agro Romano compels the proprietors within a radius of some 6 m.

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  • At Merxplas, near the Dutch frontier, is the agricultural criminal colony at which an average number of two thousand prisoners are kept employed in comparative liberty within the radius of the convict settlement.

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  • radius, ray), a metallic chemical element obtained from pitchblende, a uranium mineral, by P. and Mme.

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  • Radius >>

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  • The areas of successive surfaces vary as their radii, hence the rate of transmission Q/AT varies inversely as the radius r, and is Q/2lrrlT, if 1 is the length of the cylinder, and Q the total heat, calculated from the condensation of steam observed in a time T.

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  • The vascular bundles of the stem belong to the col xylem and the bast or phloem stand side by side on the same radius.

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  • The forests stretch on all sides within a radius of 75 m.

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  • Under favourable conditions four concentric rings may be seen round the shadow of the observer's head, the outermost, which seldom appears, having an angular radius of 40°.

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  • In old individuals the bones of the forearm (radius and ulna) became welded together about half-way down, although they remained free above.

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  • Moreover, the crowns of the hinder cheek-teeth are taller, and more distinctly crescentic, both feet are two-toed, the ulna and radius are fused, and the fibula is represented only by its lower part.

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  • Radius and ulna typically avine, 2.1 in.

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  • Not only is more powerful machinery required for the latter, but in bending it allowance has to be made for the difference in radius of outer and inner layers, which increases with increase of thickness.

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  • If 7r be the parallax, and R the radius of the earth's orbit, the distance of the star is R/sin ir.

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  • We should learn perhaps the distribution and luminosities of the stars within a sphere of radius sixty light years (corresponding to a parallax of about 0.05"), but of the structure of the million-fold greater system of stars, lying be y ond this limit, yet visible in our telescopes, we should learn nothing except by analogy.

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  • we describe a sphere about S with radius SP so as just to touch the boundaries of the stratum of stars, then, provided a class of stars is considered wholly or mainly included within this sphere, no concentration of stars in the galactic plane is to be expected, for the shape of the universe does not enter into the question.

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  • They are accordingly within the sphere of radius SP (fig.), and consequently are equally numerous in every direction.

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  • Taking a sphere whose radius is 560 light years (a distance about equal to that of the average ninth magnitude star), it will contain: I star giving fromloo,000 to io,000 times the light of the sun 26 stars „ 1,000 „ „ 1,000 „ 100 „ 22,000 „ „ 100 „ 10 „ „ „ 140,000 „ „ IO „ I „ 430,000, ,„ I, , 0.I, , n 650,000 „ „ 0 I „ 0.01 „ .„ Whether there is an increasing number of still less luminous stars is a disputed question.

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  • (Note that the z here occurring is only required to ensure harmony with tri-quaternions of which our present biquaternions, as also octonions, are particular cases.) The point whose position vector is Vrq i is on the axis and may be called the centre of the bi-quaternion; it is the centre of a sphere of radius Srq i with reference to which the point and plane are in the proper quaternion sense polar reciprocals, that is, the position vector of the point relative to the centre is Srg i.

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  • Sq/Vq, the product being the (radius)2, that is (Srq 1) 2.

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  • The point p=Vt may be called the centre of Q and the length St may be called the radius.

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  • If Q and Q' are commutative, that is, if QQ' = Q'Q, then Q and Q' have the same centre and the same radius.

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  • have a common centre and common radius.

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  • r, Radius of curvature, formula (1).

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  • The population of the city proper was 39,240 in 1901; of the city and suburbs within a to-miles radius, 162,261.

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  • Guinea-pigs placed in plague-infected houses do not contract plague if they are protected from fleas; those placed in cages protected by a border of sticky paper at least six inches in radius, which the fleas cannot jump over, do not contract plague; the others not similarly protected, do.

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  • Imagine two spheres of equal radius with 0 as their common centre, one fixed in the body and moving with it, the other fixed in space.

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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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  • P we have (T + T) sin ai,L, or T4~, or Ts/p, where p is the radius of curvature.

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    0
  • ~ the inclination to the horizontal at A or B, we have 2T~=W, AB =2p~t, approximately, where p is the radius of curvature.

    0
    0
  • Again, the mass-centre of a uniform solid right circular cone divides the axis in the ratio 3: I; that of a uniform solid hemisphere divides the axial radius in the ratio 3: 5.

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    0
  • In the case of an axial moment, the square root of the resulting mean square is called the radius of gyration of the system about the axis in question.

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    0
  • For a uniform thin circular plate, we find, taking the origin at its centre, and the axis of z normal to its plane, I~ = 1/2Maf, where M is the mass and a the radius.

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  • Since I~=Ii., I~=o, we deduce 100=3/4Ma2, ~ =4MaZ; hence the value of the squared radius of gyration isfora diameter 3/4ai, and for the axis of symmetry 3/4af.

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  • the square of the radius of gyration with respect to a diameter is ia2.

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    0
  • 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 we construct the quadric Axi+By2+Czi 2Fyz2Gzx 2HXy = M~4, (3c~) where e is an arbitrary linear magnitude, the intercept r which it makes on a radius drawn in the direction X, u, v is found by putting x, y, z=Ar, ur, Pr.

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  • The moment of inertia about any radius of the quadric (39) therefore varies inversely as the square of the length of this radius.

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  • It possesses thi property that the radius of gyration about any diameter is half thi distance between the two tangents which are parallel to that diameter, In the case of a uniform triangular plate it may be shown that thi momental ellipse at G is concentric, similar and similarly situatec to the ellipse which touches the sides of the triangle at their middle points.

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  • If k be the radius of gyration about p we find k2 =2Xarea AHEDCBAXONap, where a$ is the line in the force-diagram which represents the sum of the masses, and ON is the distance of the pole 0 from this line.

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  • If we imagine a point Q to describe a circle of radius a _________________ with the angular velocity ~, its A - 0 P orthogonal projection P on a fixed diameter AA will execute a vibration of this character.

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  • In the case of a particle falling directly towards the earth from rest at a very great distance we have C=o and, by Newtons Law of Gravitation, p/ai=g, where a is the earths radius.

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  • This may be compared with the period of revolution in a circular orbit of radius c about the same centre of force, viz.

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