• Atlanta was laid out in the form of a circle, the radius being 14 m.
• A semicircle is then drawn behind the bowls with a radius of 9 ft.
• The distance of the lucid points was the tangent of the magnified angles subtended by the stars to a radius of io ft.
• 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.
• The chief difference between the first three types lies in the weight of rails and rolling stock and in the radius of the curves.
• - 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.
• 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.
• 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.
• 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.
• 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.
• (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.
• We will now investigate the total illumination distributed over the area of the circle of radius r.
• 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).
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• Let P, P' be two consecutive positions of the radius vector.
• 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).
• 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.
• The value of C for an annular aperture of radius r and width dr is thus dC =271-Jo(Pp)pdp, (12).
• 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.
• 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.'
• 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.
• 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.
• 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.
• In the same way, he showed that the secondary bow has a greater radius than that previously assigned to it.
• 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.
• 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.
• 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 '.
• 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.
• 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.
• 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.
• 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.
• 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.
• (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.
• 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.
• 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?
• (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.
• 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.
• 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.
• 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.
• 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.
• The quadratrix of Tschirnhausen is constructed by dividing the arc and radius of a quadrant in the same number of equal parts as before.
• 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.
• 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.
• Let a solid circular sectioned cylinder of radius R 1 be enclosed in a coaxial tube of inner radius R2.
• 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.
• A census taken in July 1896 showed a population within a radius of 3 m.
• 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.
• 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.
• (X 8.) have existed since the Early U, ulna; R, radius; c, cuneiform; Eocene period.
• 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.
• 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.
• At any point a sounding line would hang in the line of the radius of curvature of the water surface.
• 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.
• 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.
• The odontoid process of the second vertebra is pig-like: and the tibia and fibula and radius and ulna are severally distinct.
• 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...
• 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.
• This device consists of a roller of radius r, pressed into contact with a disk.
• 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.
• 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.
• It extended over a circular area, with a radius of 50 m.
• 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.
• If the dispart sight were EarlyTangent being used, the sighting radius would be OD, but, as Sight.
• The formula for length of scale is, length = sighting radius X tangent of the angle of elevation.
• 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.
• 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.
• 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.
• 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.
• Denoting the constant ratio by fir, the area of a circle is ira 2, where a is the radius, and ir=3.14159 approximately.
• 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.
• 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.
• 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.
• 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.
• 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.
• In the figure the radius of the inner circle is 3 mm.
• 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.
• Then the reflected ray QR and the ray reflected at R, and so on, will all touch the circle drawn with ON as radius.
• 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.
• 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.
• 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.
• To obtain the virtual length we must add the correction for each open end, probably about I 2 radius.
• Let us suppose that the rod is circular, of radius r, and that the radial displacement of the surface is r t.
• 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.
• The radius of gyration of the section is 2a 2.
• 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.
• 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."
• 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.
• 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.
• 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.
• 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.
• Under Austria, since everywhere that 40 scholars of one nationality were to be found within a radius of 5 km.
• 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.
• 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).
• 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 %.
• 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.
• 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.
• - 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.
• Howth, Malahide and Sutton to the north, and Bray to the south, are, favoured seaside watering-places outside the radius of actual suburbs.
• The co-ordinates of its centre are - g/c, f/c; and its radius is (g 2 +f 2 - c) I.
• Since the equation to a circle of zero radius is x 2 +y 2 =o, i.e.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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."
• 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.
• 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.
• 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.
• The forests stretch on all sides within a radius of 75 m.
• 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°.
• In old individuals the bones of the forearm (radius and ulna) became welded together about half-way down, although they remained free above.
• 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.
• Radius and ulna typically avine, 2.1 in.
• 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.
• If 7r be the parallax, and R the radius of the earth's orbit, the distance of the star is R/sin ir.
• 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.
• They are accordingly within the sphere of radius SP (fig.), and consequently are equally numerous in every direction.
• 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.
• (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.
• Sq/Vq, the product being the (radius)2, that is (Srq 1) 2.
• The point p=Vt may be called the centre of Q and the length St may be called the radius.
• If Q and Q' are commutative, that is, if QQ' = Q'Q, then Q and Q' have the same centre and the same radius.
• The population of the city proper was 39,240 in 1901; of the city and suburbs within a to-miles radius, 162,261.
• 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.
• 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.
• 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.
• P we have (T + T) sin ai,L, or T4~, or Ts/p, where p is the radius of curvature.
• ~ the inclination to the horizontal at A or B, we have 2T~=W, AB =2p~t, approximately, where p is the radius of curvature.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• This may be compared with the period of revolution in a circular orbit of radius c about the same centre of force, viz.
• Tait that a similar representation of the type (30) is obtained if we replace the circle by an equiangular spiral described, with a constant angular velocity about the pole, in the direction of diminishing radius vector.
• Hence equal areas are swept over by the radius vector in equal times.
• A point on a central orbit where the radial velocity (drfdt) vanishes is called an apse, and the corresponding radius is called an apse-line.
• In elliptic harmonic motion the velocity of P is parallel and proportional to the semi-diameter CD which is conjugate to the radius CP; the hodograph is therefore an ellipse similar to the actual orbit.
• If M be the total mass, k the radius of gyration (~ ii) about the axis, we have sin 0, (3)
• If K be the radius of gyration about a parallel axis through G, we have kf=K2+h2 by If (16), and therefore i=h+K1/h, whence GO.GP=K2.
• If a be the radius of the cylinder, h the distance of G
• 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.
• The centre 0 of the disk is supposed to describe a horizontal circle of Mg~ j~ radius c with the constant angular II velocity, &, whilst its plane pre II serves a constant inclination 0 to 7/ the horizontal.
• If the particle describes a horizontal circle of angular radius a with constant angular velocity f~, we have 0=0, h=c~ sin a, and therefore f~f-cosa, (25)
• Let the angular velocity of the rotation be denoted by a=dO/dt, then the linear velocity of any point A at the distance r from the axis is or; and the path of that point is a circle of the radius r described about the axis.
• The path of a point P in or attached to the rolling cone is a spherical epitrochoid traced on the surface of a sphere of the radius OP. From P draw PQ perpendicular to the instantaneous axis.
• Let V5 denote the velocity of advance at a given instant, which of course is common to all the particles of the body; a the angular velocity of the rotation at the same instant; 2,r = 6.2832 nearly, the circumference of a circle of the radius unity.
• The radius of the pitch-circle of a wheel is called the geometrical radius; a circle touching the ends of the teeth is called the addendum circle, and its radius the real radius; the difference between these radii, being the projection of the teeth beyond the pitch-surface, is called the addendum.
• Hence the velocity of sliding is that due to this rotation about I, with the radius IT; that is to say, its value is (ai+ai).IT; (26)
• Consequently, one of the forms suitable for the teeth of wheels is the involute of a circle; and the obliquity of the action of such teeth is the angle whose cosine is the ratio of the radius of their base-circle to that of the pitch-circle of the wheel.
• The path of contact which it traces is identical with itself; and the flanks of the teeth c are internal and their faces ex ternal epicycloids for wheels, and both flanks and faces are cycloids For a pitch-circle of twice the P, - / radius of the rolling or describing /, -~- circle (as it is called) the internal B ~, epicycloid is a straight line, being, / E in fact, a diameter of the pitch- circle, so that the flanks of the teeth for such a pitch-circle are planes radiating from the axis.
• Then the radius of curvature of the epicycloid at T is For an internal epicycloid, p =4r sin o~1
• Make d bf=pc, cg=pc. From f, with the radiusfa, draw the circular arc ah; from g, with the radius ga, / B
• Trundles and Pin-Wheels.If a wheel or trundle have cylindrical pins or staves for teeth, the faces of the teeth of a wheel suitable for driving it are described by first tracing external epicycloids, by rolling the pitch-circle of the pin-wheel or trundle on the pitch-circle of the driving-wheel, with the centre of a stave for a tracing-point, and then drawing curves parallel to, and within the epicycloids, at a distance from them equal to the radius of a stave.
• A belt tends to move towards that part of a pulley whose radius is greatest; pulleys for belts, therefore, are slightly swelled in the middle, in order that the belt may remain on the pulley, unless forcibly shifted.
• The effective radius, or radius of the pitch-circle of a circular pulley or drum, is equal to the real radius added to half the thickness of the connector.
• Let r1 be the radius of the large end of each, ri that of the small end, r, that of the middle; and let Ii be the sagitta, measured perpendicular to the axes, of the arc by whose revolution each of the conoids is generated, or, in other words, the bulging of the conoids in the middle of their length.
• It is evident that the moment of friction, and the work lost b~ being performed in overcoming friction, are less in a rotating pieci as the bearings are of smaller radius.
• A with the radius of that circle; that is to say, it must be a line such as PT touching the ~ smaller circle BB, whose radius is r.
• Let N be the total pressure sustained by a flat pivot of the radius r; if that pressure be uniformly distributed, which is the case when the rubbing surfaces of the pivot and its step are both true planes, the intensity of the pressure is pN/irr2 (60)
• The friction of a conical pivot exceeds that of a flat pivot of the same radius, and under the same pressure, in the proportion of the side of the cone to the radius of its base.
• Between the concave spherical surfaces of those cups is placed a steel 0 ball, being either a complete sphere or a lens having convex surfaces of a somewhat less radius p i than the concave surfaces of the cups.
• The moment of friction of this pivot is at first almost C, inappreciable from the extreme smallness of the T radius of the circles of contact of the ball and cups, but, as they wear, that radius and the moment of friction increase.
• The moment of friction of Schieles anti-friction pivot, as it is called, is equal to that of a cylindrical journal of the radius OR=PT the constant tangent, under the same pressure.
• For inside gearing, if ri be the less radius and r, the greater, !
• Let Ti be the tension of the free part of the band at that side towards which it tends to draw the pulley, or from which the pulley tends to draw it; 1, the tension of the free part at the other side; T the tension of the band at any intermediate point of its arc of contact with the pulley; 0 the ratio of the length of that arc to the radius of the pulley; do the ratio of an indefinitely small element of that arc to the radius; F=TiT2 the total friction between the band and the pulley; dF the elementary portion of that friction due to the elementary arc do; f the coefficient of friction between the materials of the band and pulley.
• It increases with the sectional area of the rope, and is inversely proportional to the radius of the curve into which it is bent.
• The following empirical formulae for the stiffness of hempen ropes have been deduced by Mono from the experiments of Coulomb: Let F be the stiffness in pounds avoirdupois; d the diameter of the rope In inches, fl = 48d2 for white ropes and 35d2 for tarred ropes; r the effectire radius of the pulley in inches; T the tension in pounds.
• The ~ force required to constrain the weight a to move in a circle, that is the de viating force, produces an equal and -~ opposite reaction on the shaft, whose X amount F is equal to the centrifugal force Wa2 rig Ib, where r is the radius of the mass centre of the weight, and - a is its angular velocity in radians per second.
• The plane through the radius of the weight containing the axis OX is railed the axial plane because it contains the forces forming the couple due to the transference of F to the reference plane.
• In drawing these polygons the magnitude of the vector of the type Wr is the product Wr, and the direction of the vector is from the shaft outwards towards the weight W, parallel to the radius r.
• For the vector representing a couple of the type War, if the masses are all on the same side of the reference plane, the direction of drawing is from the axis outwards; if the masses are some on one side of the reference plane and some on the other side, the direction of drawing is from the axis outwards towards the weight for all masses on the one side, and from the mass inwards towards the axis for all weights on the other side, drawing always parallel to the direction defined by the radius r.
• Let a small body of the weight w undergo translation in a circulai path of the radius p, with the angular velocity of deflexion a, so that the common linear velocity of all its particles is v=ap. Then the actual energy of that body is WV2/2g = Waip2/2g.
• By comparing this with the expression for the centrifugal force (wap/g), it appears that the actual energy of a revolving body is equal to the potential energy Fp/2 due to the action of the deflecting force along one-half of the radius of curvature of the path of the body.
• Let W be the weight of a flywheel, R its radius of gyration, ai its maximum, aj its minimum, and A=~1/8(a1+ai) its mean angular velocity.
• R is called the radius of gyration of the body with regard to an axi:
• Hence his measurements are all directly comparable with modern electrostatic measurements in which the unit of capacity is that of a sphere r centimetre in radius.
• In 1828 he made inquiries about a chair at Heidelberg; and in 1830 he got a shortened Latin version of his physiological theory of colours inserted in the third volume of the Scriptores ophthalmologici minores (edited by Radius).
• Let two spherical pith balls of radius r and weight W, covered with gold-leaf so as to be conducting, be suspended by parallel silk threads of length 1 so as just to touch each other.
• For infinitely distant objects the radius of the chromatic disk of confusion is proportional to the linear aperture, and independent of the focal length (vide supra," Monochromatic Aberration of the Axis Point "); and since this disk becomes the less harmful with an increasing image of a given object, or with increasing focal length, it follows that the deterioration of the image is proportional to the ratio of the aperture to the focal length, i.e.
• The first act (the Metropolitan Police Act 1829) applied to the metropolis, exclusive of the city of London, and constituted a police area having a radius of 12 m.
• In the fore-limb the clavicle and the radius and ulna are well developed, allowing of considerable freedom of motion of the fore-paw.
• If we take the axis of z normal to either surface of the film, the radius of curvature of which we suppose to be very great compared with its thickness c, and if p is the density, and x the energy of unit of mass at depth z, then o- = f o dz, (16) and e = f a xpdz,.
• Suppose that the transition from o to s is made in two equal steps, the thickness of the intermediate layer of density la being large compared to the range of the molecular forces, but small in comparison with the radius of curvature.
• The upper surface of this column is not level, so that the height of the column cannot be directly measured, but let us assume that h is the mean height of the column, that is to say, the height of a column of equal weight, but with a flat top. Then if r is the radius of the tube at the top of the column, the volume of the suspended column is 717 2 12, and its weight is 7rpgr 2 h, when p is its density and g the intensity of gravity.
• Hence the mean height to which the fluid rises is inversely as the radius of the tube.
• This expression is the same as that for the rise of a liquid in a tube, except that instead of r, the radius of the tube, we have a the distance of the plates.
• If The Surface Could Be Treated As A Cylindrical Prolongation Of The Tube (Radius A), The Pressure Would Be T/A, And The Resulting Force Acting Downwards Upon The Drop Would Amount To One Half (2Rat) Of The Direct Upward Pull Of The Tension Along The Circumference.
• For Example, In The Case Of Water Delivered From A Glass Tube, Which Is Cut Off Square And Held Vertically, A Will Be The External Radius; And It Will Be Necessary To Suppose That The Ratio Of The Internal Radius To A Is Constant, The Cases Of A Ratio Infinitely Small, Or Infinitely Near Unity, Being Included.
• If the bubble is in the form of a sphere of radius r this material surface will have an area S = 41rr 2 (I) If T be the energy corresponding to unit of area of the film the surface-energy of the whole bubble will be ST = 41rr 2 T (2) The increment of this energy corresponding to an increase of the radius from r to r-+dr is therefore TdS = 81rrTdr (3) Now this increase of energy was obtained by forcing in air at a pressure greater than the atmospheric pressure, and thus increasing the volume of the bubble.
• Let the radius of this section PR by y, and let PT, the tangent at P, make an angle a with the axis.
• (io) Now ds - sin a (II) The radius of curvature of the meridian section is ds R1= a.
• (12) d The radius of curvature of a normal section of the surface at right angles to the meridian section is equal to the part of the normal cut off by the axis, which is R2 = PN =y/ cos a (13).
• Hence the relation between the radius vector and the perpendicular on the tangent of the rolling curve must be identical with the relation between the normal PN and the ordinate PR of the traced curve.
• Now consider a portion of a cylindric film of length x terminated by two equal disks of radius r and containing a certain volume of air.
• We know that the radius of curvature of a surface of revolution in the plane normal to the meridian plane is the portion of the normal intercepted by the axis of revolution.
• The radius of curvature of a catenary is equal and opposite to the portion of the normal intercepted by the directrix of the catenary.
• If h is the height to which the liquid will rise in a capillary tube of unit radius, then the diameter of the largest orifice is 2a =3.8317 A / (2h) = 5.4188-J (h).
• Now it is shown in hydrodynamics that she velocity of propagation of waves in deep water is that acquired by a heavy body falling through half the radius of the circle whose circumference is the wave-length, or _ f_X _ ga 27rT 'I ' v2- 2r 2r pn This velocity is a minimum when X=2.7r gp' and the minimum value is v= 4 - p g For waves whose length from crest to crest is greater than X, the principal force concerned in the motion is that of gravitation.
• When the orifice is circular of radius a, the limiting value of a is 1 J' z, where z is the least root of the equation FIG.
• If, as before, the frequency be p7211, and a the radius of the sphere, we have p 2 =n(n-1)(n+2)P a3, (6) n denoting the order of the spherical harmonic by which the deviation from a spherical figure is expressed.
• To find the radius of the sphere of water which vibrates seconds, put p = 21I, T= 81, p= 1, n= 2.
• Since the circumference of a circle is proportional to its radius, it follows that if the ratio of the radii be commensurable, the curve will consist of a finite number of cusps, and ultimately return into itself.
• Leonhard Euler (Acta Petrop. 1784) showed that the same hypocycloid can be generated by circles having radii of; (a+b) rolling on a circle of radius a; and also that the hypocycloid formed when the radius of the rolling circle is greater than that of the fixed circle is the same as the epicycloid formed by the rolling of a circle whose radius is the difference of the original radii.
• If the radius of the rolling circle be one-half of the fixed circle, the hypocycloid becomes a diameter of this circle; this may be confirmed from the equation to the hypocycloid.
• In the case of the satellites it is the period relative to the radius vector from the sun.
• The radius of curvature at any point is readily deduced from the intrinsic equation and has the value p=4 cos 40, and is equal to twice the normal which is 2a cos 2B.
• The cartesian equation in terms similar to those used above is x = a6+b sin 0; y=a-b cos 0, where a is the radius of the generating circle and b the distance of the carried point from the centre of the circle.
• In both cases the curves are epicycloids; in the first case the radii of the rolling and the fixed circles are a(2n - I) /4n and a/2n, and in the second, an/(2n+ I) and a/(2n4-I), where a is the radius of the mirror and n the number of reflections.
• The Cartesian equation to the caustic produced by reflection at a circle of rays diverging from any point was obtained by Joseph Louis Lagrange; it may be expressed in theform 1(4,2_ a2) (x 2+ y2) - 2a 2 cx - a 2 c 2 1 3 = 2 7 a4c2y2 (x2 + y2 - c2)2, where a is the radius of the reflecting circle, and c the distance of the luminous point from the centre of the circle.
• We then find that the density would increase as we go outwards, at first slowly, but finally with extreme rapidity, the last tenth of the radius comprising half the mass.
• Young records one which reached an elevation of 350,000 m., or more than three-quarters of the sun's radius.
• Apart then from absorption there will be a discontinuous change in brightness in the apparent disk at that value of the angular radius d which corresponds to tangential emission from the upper lever r' of this mirage-forming region.
• The greater µ is, the greater would be the value of d, the apparent angular radius, corresponding to horizontal emission from a given level r, and that whether we accept Schmidt's theory or not.
• The latter dam is curved in plan, the radius being 740 ft.
• The extension of a spiral spring is given by the formula: Extension =W4n1VÃ† r 4, in which W = weight causing extension, in lbs; n = number of coils; R = radius of spring, from centre of coil to centre of wire, in inches; r = radius of wire of which the spring is made, in inches; E = coefficient of elasticity of wire, in lbs per square inch.
• Until 1907, when the city was enlarged by annexation, its limits remained as they were first laid out, a circle with a radius of r m., the court house being its centre.
• The magnitude of the unbalanced force, for a mass of w pounds at a radius of r feet and a velocity of v feet per second, is expressed by wv 2 /gr lb; and, since the force varies as the square of the velocity, it is necessary carefully to balance a pulley running at a high speed to prevent injurious vibrations.
• In practice the pulley rim is curved to a radius of from three to five times its breadth, and this not only guides the belt, but allows the line of direction of the advancing side to deviate to a small extent, depending on the elasticity of the material.
• The proportions of cone pulleys for open or crossed belts may be determined by considering the expression for the half length (1) of a belt wrapping round pulleys of radius r 1 and r 2 respectively, and with centres distant c apart.
• F B is the evolute of this circle, and for any radius DE at an angle a and corresponding tangent EG terminated by the evolute, the perpendicular distance of G from the line AD is c(cos a+a sin a).
• A circular arc, centre D and radius c/2, meets D E in K, and the perpendicular KL gives 2c sin a.
• It is of considerable importance that the effective radius of action of the rope remain constant throughout each pulley, otherwise the wear on the rope becomes very great and its life is diminished.
• The effect of pin friction is equivalent to diminishing the radius of the effort and increasing that of the resistance.
• Then place a Lens of about three foot radius (suppose a broad Object-glass of a three foot Telescope), at the distance of about four or five foot from thence, through which all those colours may at once be transmitted, and made by its Refraction to convene at a further distance of about ten or twelve feet.
• Johann Kepler had proved by an elaborate series of measurements that each planet revolves in an elliptical orbit round the sun, whose centre occupies one of the foci of the orbit, that the radius vector of each planet drawn from the sun describes equal areas in equal times, and that the squares of the periodic times of the planets are in the same proportion as the cubes of their mean distances from the sun.
• Any one of these is a " parallel " of the given curve; and it can be obtained as the envelope of a circle of constant radius having its centre on the given curve.
• The co-ordinates of P will then be the following three quantities: - (I) The length of the line OP, or the distance of the body from the origin, which distance is called the radius vector of the body.
• (2) The angle XOQ which the projection of the radius vector upon the fundamental plane makes with the initial line OX.
• (3) The angle QOP, which the radius vector makes with the fundamental plane.
• Imagining, as we may well do, that the radius of this sphere is infinite - then every direction, whatever the origin, may be represented by a point on its surface.
• The third law enables us to compute the time taken by the radius vector to sweep over the entire area of the orbit, which is identical with the time of revolution.
• The problem of constructing successive radii vectores, the angles of which are measured off from the radius vector of the body at the original given position, is then a geometric one, known as Kepler's problem.
• This reasoning tacitly supposes the orbit to be a circle of radius a, and the mass of the planet to be negligible.
• Taking as the radius vector of each body the line from the body to the common centre of gravity of all, the sum of the products formed by multiplying each area described, by the mass of the body, remains a constant.
• The speed of the latter may, therefore, be expressed as a function of its radius vector at the moment and of the major axis of its orbit without introducing any other elements into the expression.
• Hansen, therefore, shows how the radius vector is corrected so as to give that of the true planet.
• This plane remains invariable so long as no third body acts; when it does act the position of the plane changes very slowly, continually rotating round the radius vector of the planet as an instantaneous axis of rotation.
• Of the three co-ordinates,the radius vector does not admit of direct measurement, and must be inferred by a combination of indirect measurements and physical theories.
• Bessel announced, in December 1838, the perspective yearly shifting of 61 Cygni in an ellipse with a mean radius of about one-third of a second.
• Other numerical particulars relating to the moon are: Mean distance from the earth (earth's radius as I) ..
• The law of this motion was such that the phenomena could be represented by supposing the motion to be actually circular and uniform, the apparent variations being explained by the hypothesis that the earth was not situated in the centre of the orbit, but was displaced by an amount about equal to one-twentieth of the radius of the orbit.
• By decree of the 23rd of May 1907, the radius of the circle within which claims may be pegged is 2 kilometres (14 m.), and a tax of 5% is levied on the value of the gold extracted.
• This document fixed the frontier of the British protectorate inland at a radius of 10 m.
• The angle cp is termed the eccentric angle, and is geometrically represented as the angle between the axis of x (the major axis of the ellipse) and the radius of a point on the auxiliary circle which has the same abscissa as the point on the ellipse.
• An arc with centre 0 and radius OB forms part of a curve.
• Then with centre 0 1 and radius OJ, =OA 1, describe an arc. By reflecting the two arcs thus described over the centre the ellipse is approximately described.
• There is but one; we must assume that the first A of every series is identical, just as the centre is the same point in every radius.
• As the planet revolves around the centre, each radius vector describes a surface of which the area swept over in a unit of time measures the areal velocity of the planet.
• The shaft gradually tapers below and is firmly welded to the radius.
• The pisiform is large and prominent, flattened and curved; it articulates partly with the cuneiform and partly with the lower end of the radius.
• He decided to clear the district of rebels within a radius of 30 m.
• Radius confluent with ulna, and tibia with fibula; tarsus (astragalus and calcaneum) elongate, forming an additional segment in the hind limb.
• The orbit is surrounded by a bony ring; the ulna and radius in the fore, and the tibia and fibula in the hind-limb are united, and the feet are of the types described above.
• He called to muster everyone in ear shot to inaugurate a building to building search, beginning in a ten mile radius of the abandoned van.
• Gerry, the station chief, tripped his wards a moment before the stealthy Guardian crossed the threshold into the ten meter radius around Xander, where he was able to absorb thoughts and manipulate minds.
• Synchrotron A type of circular accelerator in which the particles travel in synchronized bunches at fixed radius.
• Thus an INNER radius of 1 means that the sky annulus starts where the object aperture ends.
• Most of the tourist attractions lie within a mile's radius of the famous Charles Bridge.
• The white metal is then machined to the correct radius using the big borer.
• I also have a bowing in the left arm where the radius is shorter than the ulna bone.
• The entire electromagnetic calorimeter at TESLA comprises a cylinder of length 5.5m, internal radius 1.9m, and annular thickness of 20cm.
• There were no significant differences in cortical volumetric bone mineral density (vBMD) at the radius or tibia diaphysis between the groups.
• Struck by the loader bucket if the access platform is within the working radius of the loading machine (for 360 degree excavators ).
• Power series, radius of convergence, important examples including exponential, sine and cosine series.
• Firstly, the single-piece neck has a 12-inch radius fingerboard, which is flatter than either American or American Vintage specs.
• Has normal frag grenades + Emp grenades which can kill multiple targets in a large radius.
• The grenade, tho less powerful than a standard fragmentation grenade, has a lethal radius of 5m.
• In its industrial heyday 90% of Britain's copper smelting capacity was located within a 20 mile radius of Swansea.
• In this case it is the radius of the hydrogen-bonding hydrogen-bonding hydrogens which is reduced, rather than the radius of the central nitrogen itself.
• Examples of these conditions are fractures of the distal radius, hand phalanxes, tibial diaphysis, proximal femur and proximal humerus.
• For a circle of radius r a circle of radius r we wish to locate it so that it smoothly joins the straight sections of the corner.
• The automatic choice of the cut-off radius for RF is twice the radius of gyration.
• The bend radius is set at 24 turns for the cylindrical mirror.
• The project must be within a 20-mile radius of a B&Q store.
• Dr. Chris Busby, EU child health expert, explained: ' Small particles will move within a 10-mile radius.
• The type of atomic radius being measured here is called the metallic radius or the covalent radius depending on the bonding.
• There are around 10 licensed companies within a 50-mile radius of Cleveland.
• Atomic and ionic radius increase down both groups as can be predicted from the increasing number of shells.
• In 2003, Waitrose launched a " locally produced " range where the food sold in a store is farmed within a 30-mile radius.
• Schwarzschild radius The radius r of the event horizon for a Schwarzschild black hole.
• Remember 80,000 of those people live within a five-mile radius of the hospital.
• The project area extends to a four mile radius from the center of Norwich to the fringe.
• A compromise was reached whereby only cases discovered within a one-mile radius of the hospital would be admitted.
• Nearby Attractions Two great Lakeland pubs lie within a two-mile radius of the site providing excellent pub food.
• Its aim is to provide a friendly service for the town, and people living within a ten-mile radius.
• A variety of dining options are within a three-mile radius of the hotel.
• Heh, I know the Indian restaurants within a three mile radius of Finchley better than I know the pubs.
• A line is drawn from the bicipital tuberosity to the most ulnar aspect of the radius at the wrist.
• Leave well alone but can do shortening of distal ulna and radius closing wedge osteotomy.
• In order to find the whole emission of energy from one particle (T), we have to integrate the square of (3) over the surface of a sphere of radius r.
• We learn that the light dispersed in the direction of primary vibration is not only of higher order in the difference of optical quality, but is also of order k 2 c 2 in comparison with that dispersed in other directions, where c is the radius of the sphere, and k=21r/X as before.
• On the other hand, when the ancestral condition is modified, it may be regarded as having moved outwards along some radius from the archecentric condition.
• The radius is the straighter and more slender of the two forearm bones.
• P is the position of the planet at any time, and we call r the radius vector FP. The angle AFP between the pericentre and the position P of the planet is the anomaly called v.
• The true anomaly, AFP, is commonly determined through the mean anomaly conceived thus: Describe a circle of radius a= CA around F, and let a fictitious planet start from K at the same moment that the actual planet passes A, and let it move with a uniform speed such that it shall complete its revolution in the same time T as the actual planet.
• It is there shown that the surface tension of a liquid may be calculated from its rise in a capillary tube by the formula y = rhs, where y is the surface tension per square centimetre, r the radius of the tube, h the height of the liquid column, and s the difference between the densities of the liquid and its vapour.
• (II) A single vortex in a circular cylinder of radius a at a distance c from the centre will move with the velocity due to an equal opposite image at a distance a 2 /c, and so describe a circle with velocity mc/(a 2 -c 2)in the periodic time 21r(a 2 -c 2)/m.
• Considered by itself, with the cylinders held fixed, the vortex sets up a circumferential velocity m/r on a radius r, so that the angular momentum of a circular filament of annular cross section dA is pmdA, and of the whole vortex is pm7r(b2-a2).
• Thus the capacity of a sphere in electrostatic units (E.S.U.) is the same as the number denoting its radius in centimetres.
• But the great stretch of highly irrigated and valuable fruit-growing land, which appears to spread from the walls of Herat east and west as far as the eye can reach, and to sweep to the foot of the hills north and south with an endless array of vineyards and melon-beds, orchards and villages, varied with a brilliant patchwork of poppy growth brightening the width of green wheat-fields with splashes of scarlet and purple - all this is really comprised within a narrow area which does not extend beyond a ten-miles' radius from the city.
• If a is the radius of a sphere, then (i) volume of sphere =tira3; (ii) surface of sphere=41ra 2 =curved surface of circumscribing cylinder.
• He showed that the surface of a segment is equal to the area of the circle whose radius equals the distance from the vertex to the base of the segment; that the surface of the entire sphere is equal to the curved surface of the circumscribing cylinder, and to four times the area of a great circle of the sphere; and that the volume is twothirds that of the circumscribing cylinder.
• For a projectile in which the ogival head is struck with a radius of 2 diameters, Bashforth puts K= o 975; on the other hand, for a flat-headed projectile, as required at proof-butts, = 1 .
• Diagram of the structure of a medusa; the ectoderm is left clear, the endoderm is dotted, the mesogloea is shaded black; a-b, principal axis (see Hydrozoa); to the left of this line the section is supposed to pass through an inter-radius (I.R.); to the right through a radius (R).
• Taking the circumference as intermediate between the perimeters of the inscribed and the circumscribed regular n-gons, he showed that, the radius of the circle being given and the perimeter of some particular circumscribed regular polygon obtainable, the perimeter of the circumscribed regular polygon of double the number of sides could be calculated; that the like was true of the inscribed polygons; and that consequently a means was thus afforded of approximating to the circumference of the circle.
• 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 40Ã‚°.
• For instance, considering four equal particles at the vertices of a regular tetrahedron, we can infer that the radius R of the circumscribing sphere is given by R2=j a2, if a be the length of an edge.
• If the pencil with the angle u 2 be that of the maximum aberration of all the pencils transmitted, then in a plane perpendicular to the axis at O' 1 there is a circular " disk of confusion" of radius 0' 1 R, and in a parallel plane at 0'2 another one of radius 0' 2 R 2; between these two is situated the " disk of least confusion."
• (4) After eliminating the aberration on the axis, coma and astigmatism, the relation for the flatness of the field in the third order is expressed by the " Petzval equation," I 1 = o, where is the radius of a refracting surface, n and n' the refractive indices of the neighbouring media, and / the sign of summation for all refracting surfaces.
• The catenaries which lie between the two whose direction coincides with the axis of revolution generate surfaces whose radius of curvature convex towards the axis in the meridian plane is less than the radius of concave curvature.
• The catenaries which lie beyond the two generate surfaces whose radius of curvature convex towards the axis in the meridian plane is greater than the radius of concave curvature.
• The extension of a spiral spring is given by the formula: Extension =W4n1VÃƒâ€  r 4, in which W = weight causing extension, in lbs; n = number of coils; R = radius of spring, from centre of coil to centre of wire, in inches; r = radius of wire of which the spring is made, in inches; E = coefficient of elasticity of wire, in lbs per square inch.
• If F be the number of faces, n the number of edges per face, m the number of faces per vertex, and l the length of an edge, and if we denote the angle between two adjacent faces by I, the area by A, the volume by V, the radius of the circum-sphere by R, and of the in-sphere by r, the following general formulae hold, a being written for 21r/n, and a for 27r/m:- Sin z I =cos 1 3/sin a; tan II =cos l3/ (sin'- a -cos t R) 2.
• For a circle of radius r we wish to locate it so that it smoothly joins the straight sections of the corner.
• A shaving or make-up mirror of this type has a radius of curvature of 30 cm.
• Post refinement gives very accurate cell parameters but has a relatively small radius of convergence.
• Here b is the radius of a three sphere of constant distance, sigma, from the north pole of the instanton.
• Radius Topology works seamlessly with MapInfo products in an Oracle environment.
• With every step, the radius of the circular component of your path produced by the rotation of the earth gets larger.
• The grating substrates have a sagittal radius of 60 cm.
• The maximum shear stress is given by the radius of the largest circle.
• A sphere of 200 megaparsecs radius centered on the earth is large enough to contain several large clusters of galaxies.
• An egg has an equator which is smaller than that of a sphere with the same polar radius.