# Radii Sentence Examples

Circles of these

**radii**are usually marked around the jack for convenience' sake.The resistance to motion round a curve has not been so systematically studied that any definite rule can be formulated applicable to all classes of rolling stock and all

**radii**of curves.Thus, while halos have certain definite

**radii**, viz.Let v be the common velocity of the two pitch-circles, ri, C2, their

**radii**; then the above equation becomes /1 IIt may be readily shown that the external and internal centres are the points where the line joining the centres of the two circles is divided externally and internally in the ratio of their

**radii**.The chief difficulty in this method lay in determining the effective distances of the bulbs of the thermometers from the axis of the cylinder, and in ensuring uniformity of flow of heat along different

**radii**.The angular velocities of a pair of connected circular pulleys or drums are inversely as the effective

**radii**.The aberrations can also be expressed by means of the "characteristic function " of the system and its differential coefficients, instead of by the

**radii**, &c., of the lenses; these formulae are not immediately applicable, but give, however, the relation between the number of aberrations and the order.The

**radii**, thicknesses and distances are continually altered until the errors of the image become sufficiently small.From this he showed that the rise of the liquid in tubes of the same substance is inversely proportional to their

**radii**.AdvertisementHe thus found for the pressure at a point in the interior of the fluid an expression of the form p =K+ZH(1/R+i/R'), where K is a constant pressure, probably very large, which, however, does not influence capillary phenomena, and therefore cannot be determined from observation of such phenomena; H is another constant on which all capillary phenomena depend; and R and R' are the

**radii**of curvature of any two normal sections of the surface at right angles to each other.Since e is a line of insensible magnitude compared with the dimensions of the mass of liquid and the principal

**radii**of curvature of its surface, the volume of the shell whose surface is S and thickness will be and that of the interior space will be V - SE.Let us examine the case in which the particle m is placed at a distance z from a curved stratum of the substance, whose principal

**radii**of curvature are R 1 and R2.When the surface is curved, the effect of the surface-tension is to make the pressure on the concave side exceed the pressure on the convex side by T (1 /R I i /R 2), where T is the intensity of the surface-tension and R 1, R2 are the

**radii**of curvature of any two sections normal to the surface and to each other.This equation, which gives the pressure in terms of the principal

**radii**of curvature, though here proved only in the case of a surface of revolution, must be true of all surfaces.AdvertisementI or the curvature of any surface at a given point may be completely defined in terms of the positions of its principal normal sections and their

**radii**of curvature.Hence a catenoid whose directrix coincides with the axis of revolution has at every point its principal

**radii**of curvature equal and opposite, so that the mean curvature of the surface is zero.Their sexual cells are (probably in all cases) produced from the ectoderm, and lie in those

**radii**which are first accentuated in development.The sexual cells of the medusoid lie in the endoderm on interradii, that is, on the second set of

**radii**accentuated in the course of development.The sexual cells are borne on the mesenteries in positions irrespective of obvious developmental

**radii**.AdvertisementIn the annexed figure, there are shown various examples of the curves named above, when the

**radii**of the rolling and fixed circles are in the ratio of I to 3.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.In the particular case when the

**radii**are in the ratio of I to 3 the epicycloid (curve a) will consist of three cusps external to the circle and placed at equal distances along its circumference.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**.The epicychid when the

**radii**of the circles are equal is the cardioid (q.v), and the corresponding trochoidal curves are limacons.AdvertisementIn 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.In determining the dimensions of corresponding drums of cone pulleys it is evident that for a crossed belt the sum of the

**radii**of each pair remains a constant, since the angle a is constant, while for an open belt a is variable and the values of the**radii**are then obtained by solving the equations r 1 = l/ir - c(a sin a + cos a) + 2c sin a, r 2 = l/7r - c(a sin a +cos_a) - lc sin a.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.Although the longitude of the fictitious planet at the fictitious time is then equal to that of the true planet at the true time, their

**radii**vectores will not be strictly equal.Two polyhedra correspond when the

**radii**vectores from their centres to the mid-point of the edges, centre of the faces, and to the vertices, can be brought into coincidence.With increase of speeds this matter has become important as an element of comfort in passenger traffic. As a first approximation, the centre-line of a railway may be plotted out as a number of portions of circles, with intervening straight tangents connecting them, when the abruptness of the changes of direction will depend on the

**radii**of the circular portions.Representing by P this position, it follows that the area of that portion of the ellipse contained between the

**radii**vectores FB and FP will bear the same ratio to the whole area of the ellipse that t does to T, the time of revolution.The potential due to a single pole of strength m at the distance r from the pole is V = m/ r, (7) the equipotential surfaces being spheres of which the pole is the centre and the lines of force

**radii**.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.Primary, secondary and spurious bows were formed, and their

**radii**measured; a comparison of these observations exhibited agreement with Airy's analytical values.This will be evident if we consider that, since

**radii**vectores of the hodograph represent velocities in the orbit, the elementary arc between two consecutive**radii**vectores of the hodograph represents the velocity which must be compounded with the velocity of the moving point at the beginning of any short interval of time to get the velocity at the end of that interval, that is to say, represents the change of velocity for that interval.**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.For figures of more than four sides this method is not usually convenient, except for such special cases as that of a regular polygon, which can be divided into triangles C by

**radii**drawn from its centre.The solid enclosed by a small circle and the

**radii**vectores from the centre of the sphere is a "spherical sector"; and the solid contained between two spherical sectors standing on copolar small circles is a "spherical cone."If r, r i be the

**radii**of two spheres, d the distance between the centres, and 0 the angle at which they intersect, then d2=r2+ r12 2rr l cos ¢ hence 2rr 1 cos =d2r2 - r22.Divide the span L into any convenient number n of equal parts of length 1, so that nl = L; compute the

**radii**of curvature R 1, R2, R3 for the several sections.Let measurements along the beam be represented according to any convenient scale, so that calling L 1 and 1 1 the lengths to be drawn on paper, we have L = aL i; now let r1, r 2, r 3 be a series of

**radii**such that r 1 = R i /ab, r 2 = R 2 /ab, &c., where b is any convenient constant chosen of such magnitude as will allow arcs with the**radii**, r 1, &c., to be drawn with the means at the draughtsman's disposal.The medusa has a pronounced radial symmetry, and the positions of the primary tentacles, usually four in number, mark out the so-called

**radii**, alternating with which are four interradii.Starting with the stem forms the descendants of which have passed through either persistent or changed habitats, we reach the underlying idea of the branching law of Lamarck or the law of divergence of Darwin, and find it perhaps most clearly expressed in the words "adaptive radiation" (Osborn), which convey the idea of

**radii**in many directions.Among extinct Tertiary mammals we can actually trace the giving off of these

**radii**in all directions, for taking advantage of every possibility to secure food, to escape enemies and to reproduce kind; further, among such well-known quadrupeds as the horses, rhinoceroses and titanotheres, the modifications involved in these radiations can be clearly traced.Because of the repetition of analogous physiographic and climatic conditions in regions widely separated both in time and in space, we discover that continental and local adaptive radiations result in the creation of analogous groups of

**radii**among all the vertebrates and invertebrates.The figure included by two

**radii**and an arc is a " sector," e.g.It may be shown to be the locus of the vertex of the triangle which has for its base the distance between the centres of the circles and the ratio of the remaining sides equal to the ratio of the

**radii**of the two circles.Then circles having the intersections of tangents to this circle and the line of centres for centres, and the lengths of the tangents as

**radii**, are members of the coaxal system.The remiges and rectrices indicate perfect feathers, with shaft and complete vanes which were so neatly finished that they must have possessed typical

**radii**and hooklets.It has now been firmly established, both experimentally and mathematically, that coronae are due to diffraction by the minute particles of moisture and dust suspended in the atmosphere, and the

**radii**of the rings depend on the size of the diffracting particles.The law, e.g., of the equality of the

**radii**of a circle cannot be exhibited to sense, even if equal**radii**may be so exhibited.This makes the Euphrates the main eastern limit, with

**radii**to the north-east angle of the Levant and the south-east angle of the Black Sea, and roughly agrees with the popular conception of Asia Minor as a geographical region.The squares of the

**radii**of gyration about the principal axes at P may be denoted by k,i+k32, k,f + ki2, k12 + k,2 hence by (32) and (35), they are rfOi, r2Oi, r20s, respectively.In the wheel and axle, motion is received and transmitted by two cylindrical surfaces of different

**radii**described about their common fixed axis of turning, their velocity-ratio being that of their**radii**.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, in any pair of circular wheels which work together, the numbers of teeth in a complete circumference are directly as the

**radii**and inversely as the angular velocities.Set off ab = ac = 1/2p. Draw

**radii**bd, Ce; draw fb, cg, making angles of e 753/4 with those**radii**.The length L of an endless belt connecting a pair of pulleys whose effective

**radii**are r,, r,, with parallel axes whose distance apart is c, is given by the following formulae, in each of which the first term, containing the radical, expresses the length of the straight parts of the belt, and the remainder of the formula the length of the curved parts.The speed-cones are either continuous cones or conoids, as A, B, whose velocity ratio can be varied gradually while they are in motion by shifting the belt, or sets of pulleys whose

**radii**vary by steps, as C, D, in which case the velocity ratio can be changed by shifting the belt from one pair of pulleys to another.He investigated the optical constants of the eye, measured by his invention, the ophthalmometer, the

**radii**of curvature of the crystalline lens for near and far vision, explained the mechanism of accommodation by which the eye can focus within certain limits, discussed the phenomena of colour vision, and gave a luminous account of the movements of the eyeballs so as to secure single vision with two eyes.Integrating with respect to f from f =z to f=a, where a is a line very great compared with the extreme range of the molecular force, but very small compared with either of the

**radii**of curvature, we obtain for the work (1,G (z) - 111(a))dw, and since (a) is an insensible quantity we may omit it.If the ratio of the

**radii**be as I to 4, we obtain the four-cusped hypocycloid, which has the simple cartesian equation x 2'3+ y 213 = a 21 '.The longitudes, latitudes and

**radii**vectores of a planet, being algebraically expressed as the sum of an infinite periodic series of the kind we have been describing, it follows that the problem of finding their co-ordinates at any moment is solved by computing these expressions.Suppose that from the centre of gravity of the solar system (instead of which we may, if we choose, take the centre of the sun), lines or

**radii**vectores be drawn to every body of the solar system.By selecting the

**radii**of the surfaces and the kind of glass the underor over-correction can be regulated.When the axis is so shortened that the secondary axes arise from a common point, and spread out as

**radii**of nearly equal length, each ending in a single flower or dividing again in a similar radiating manner, an umbel is produced, as in fig.Unlike LASCO, it will make measurements and images between 2 and 10 solar

**radii**with high spectral and spatial resolution.Propositions I-II are preliminary, 13-20 contain tangential properties of the curve now known as the spiral of Archimedes, and 21-28 show how to express the area included between any portion of the curve and the

**radii**vectores to its extremities.**Radii**a multiple of four, with radial gastric pouches bifurcated or subdivided; the tentacles are implanted in the notch between the two subdivisions of each (primary) gastric pouch, hence the (secondary) gastric pouches appear to be " internemal " in position, i.e.A "spherical sector" and "spherical cone" may be also regarded as the solids of revolution of a circular sector about one of its bounding

**radii**, and about any other line through the vertex respectively.The curve may be regarded as an epitrochoid (see Epicycloid) in which the rolling and fixed circles have equal

**radii**.Furthermore it is seen that AB is perpendicular to the line joining the centres, and divides it in the ratio of the squares of the

**radii**.This is the only way, we say; but there are as many ways as there can be drawn

**radii**from one centre.It is the envelope of circles described on the central

**radii**of an ellipse as diameters.The probable reason for the wall-lines being concentric is that lines passing over the

**radii**as nearly as possible at right angles are the shortest that can be laid on; they therefore use up a smaller quantity of silk and take a shorter time to spin than threads crossing the**radii**in any other direction; and at the same time they afford them the greatest possible support compatible with delicacy and strength of construction.