 • 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.

• It is the envelope of circles described on the central radii of an ellipse as diameters.

• Thus, while halos have certain definite radii, viz.

• The figure included by two radii and an arc is a " sector," e.g.

• 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 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.

• 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.

• the inner halo I, and the outer halo 0, having radii of about 22° and 46 °, and exhibiting the colours of the spectrum in a confused manner, the only decided tint being the red on the inside.

• Q and KQ have a common centre and equal and opposite radii; that is, the t of KQ is the negative conjugate of that of Q.

• Set off ab = ac = 1/2p. Draw radii bd, Ce; draw fb, cg, making angles of e 753/4 with those radii.

• Let v be the common velocity of the two pitch-circles, ri, C2, their radii; then the above equation becomes /1 I

• the radii, thicknesses, refractive indices and distances between the lenses, was solved by L.

• 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.

• 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.

• Q and KQ have a common centre and equal and opposite radii; that is, the t of KQ is the negative conjugate of that of Q.

• Set off ab = ac = 1/2p. Draw radii bd, Ce; draw fb, cg, making angles of e 753/4 with those radii.

• 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.

• 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.

• 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.

• in diameter, sometimes surmounted by trees in the midst of a treeless plain and sometimes arranged in circles and on radii, and decreasing in size with distance from the centre of the field - has been variously explained.

• 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 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.

• 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.

• and the path of such a point is a circle described about A with the radius AB, being for outside rolling the sum, and for inside rolling the difference, of the radii of the cylinders.

• Then, for every possible pair of points of contact, the two following equations must be simultaneously fulfilled: Sum of radii, CiUi+C2U2==C1T,+C,Ti = constant; arc, T2U2 = TiUi.

• 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.

• Draw CiP1, C2P2 perpendicular to P~IP2, and with those lines as radii describe about the centres of the wheels the,circles DiD1, D2D2, called base-circles.

• It is evident that the radii of the base-circles bear to each other the same proportions as the radii of the pitch-circles, and also that CiPi=1C1.cos obliquity ~ (27)

• 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.

• _______) Now c is constant because the ________ - axes are parallel; therefore the sum of the radii of the pitch ~ circles connected in every - position of the belt is to be ~ constant.

• The radii at the middle and end being thus ~ietermined, make the generating curve an arc either of a circle or of a parabola.

• 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.

• 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.

• 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.

• 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.

• There are thus marked out a series of circles, whose radii x are given by x 2 +r = (r-{- 2nX) 2, or x = nar nearly; so that P ?

• Snellius) to calculate the radii of the bows, and his theoretical angles were in agreement with those observed.

• 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.

• 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.

• 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.

• 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 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.

• 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.

• 4) in which the field-lens is changed into a meniscus having radii in about the ratio of +I to - 9 gives still better results, but still not quite so good as the results obtained by using the combination of two convexo-plane lenses of the focal ratio 2 to I.

• The speed is very nearly four radii of the earth's orbit per year; thus the annual parallactic motion is equal to four times the parallax, for a star lying in a direction 90° from the solar apex; for stars nearer the apex or antapex it is foreshortened.

• 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.

• where p, p are the radii of curvature of the two curves at J, 4~ is the inclination of the common tangent at J to the horizontal, and h is the height of G above J.

• 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.

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

• If ~s be an element of the path, p3s is twice the area enclosed by s and the radii drawn to its extremities from 0.

• 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.

• But a limit is put to th~ diminution of the radii of journals and pivots by the conditions 01 durability and of proper lubrication, and also by conditions 01 strength and stiffness.

• 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.

• Snellius) to calculate the radii of the bows, and his theoretical angles were in agreement with those observed.

• 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."

• 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.

• 4) in which the field-lens is changed into a meniscus having radii in about the ratio of +I to - 9 gives still better results, but still not quite so good as the results obtained by using the combination of two convexo-plane lenses of the focal ratio 2 to I.

• If ~s be an element of the path, p3s is twice the area enclosed by s and the radii drawn to its extremities from 0.

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

• 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.

• 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.

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

• 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.

• It 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.

• For a single lens of very small thickness and given power, the aberration depends upon the ratio of the radii r: r', and is a minimum (but never zero) for a certain value of this ratio; it varies inversely with the refractive index (the power of the lens remaining constant).

• The radii, thicknesses and distances are continually altered until the errors of the image become sufficiently small.

• These constants are determined by the data of the system (radii, thicknesses, distances, indices, &c., of the lenses); therefore their dependence on the refractive index, and consequently on the colour, are calculable (the formulae are given in Czapski-Eppenstein, G'rundziige der Theorie der optischen Instrumente (1903, p. 166).

• (b) Two thin lenses in contact: let 4) i and 4) 2 be the powers corresponding to the lenses of refractive indices n i and n 2 and radii r' 1, r" i, and r' 2, r" 2 respectively; let 4) denote the total power, and dc/), 2 the changes of 4), n l, and n 2 with the colour.

• Io); the values of the four radii must satisfy the equations (2) and (4).

• From this he showed that the rise of the liquid in tubes of the same substance is inversely proportional to their radii.

• He 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.

• 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.

• We may also write ur 1 = I +zu 1+ &c., since z is very small compared with u, and expressing u in terms of w by (25), (we find l 21- mv i fi(z) i I +z(c R w + ' R 2 w) do) = 27rmoti(z) I -f-ZZ (Ki + R2/ This then expresses the work done by the attractive forces when a particle m is brought from an infinite distance to the point P at a distance z from a stratum whose surface-density is a, and 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.

• I 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.

• In 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.

• the original position of the moving point on the fixed circle), a and b being the radii of the fixed and rolling circles, and 0 the angle through which the line joining the centres of the two circles has passed.

• 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 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 epicychid when the radii of the circles are equal is the cardioid (q.v), and the corresponding trochoidal curves are limacons.

• an epicycloid in which the radii of the fixed and rolling circles are equal.

• 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.

• 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.

• 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.

• Draw any number of radii of the two circles, and from the points of intersection with the major circle draw lines parallel to the minor axis, and from the points of intersection with the minor circle draw lines parallel to the major axis.

• 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.

• (Reference should be made to the article Crystallography for illustrations, and for applications of these phenomena to the determination of crystal form.) With an uniaxal plate perpendicular to the optic axis, the curves of constant retardation are concentric circles and the lines of like polarization are the radii: thus with polarizer and analyser regulated for extinction, the pattern consists of a series of bright and dark circles interrupted by a black cross with its arms parallel to the planes of polarization and analysation.

• 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.

• blast radii from Bazooka to HHG standard.

• ellipsoid defined by the error radii centered on P1.

• ionic radii taken from the Nuffield Advanced Science Book of Data.

• lytic lesions in the cortex of both radii.

• 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.

• Circles of these radii are usually marked around the jack for convenience' sake.

• 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.

• 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.

• 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.

• 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.

• 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.

• in diameter, sometimes surmounted by trees in the midst of a treeless plain and sometimes arranged in circles and on radii, and decreasing in size with distance from the centre of the field - has been variously explained.

• 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.

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

• There are thus marked out a series of circles, whose radii x are given by x 2 +r = (r-{- 2nX) 2, or x = nar nearly; so that P ?

• 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.

• Let 0 be the common centre of two circles, of radii a and b, and let radii enclosing an angle 0 (circular measure) cut their circumferences in A, B and C, D respectively (fig.

• 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.

• 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.

• 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.

• 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 curve may be regarded as an epitrochoid (see Epicycloid) in which the rolling and fixed circles have equal radii.

• The figure included by two radii and an arc is a " sector," e.g.

• It 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.

• 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.

• 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.

• 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 inner halo I, and the outer halo 0, having radii of about 22Ã‚° and 46 Ã‚°, and exhibiting the colours of the spectrum in a confused manner, the only decided tint being the red on the inside.

• 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 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.

• Thus, while halos have certain definite radii, viz.

• 22Ã‚° and 46Ã‚°, the radii of coronae vary very considerably; also, halos are coloured red on the inside, whereas coronae are coloured red on the outside '(see' Halo) .

• 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 speed is very nearly four radii of the earth's orbit per year; thus the annual parallactic motion is equal to four times the parallax, for a star lying in a direction 90Ã‚° from the solar apex; for stars nearer the apex or antapex it is foreshortened.

• 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.

• where p, p are the radii of curvature of the two curves at J, 4~ is the inclination of the common tangent at J to the horizontal, and h is the height of G above J.

• 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.

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

• 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.

• and the path of such a point is a circle described about A with the radius AB, being for outside rolling the sum, and for inside rolling the difference, of the radii of the cylinders.

• Then, for every possible pair of points of contact, the two following equations must be simultaneously fulfilled: Sum of radii, CiUi+C2U2==C1T,+C,Ti = constant; arc, T2U2 = TiUi.

• 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.

• Draw CiP1, C2P2 perpendicular to P~IP2, and with those lines as radii describe about the centres of the wheels the,circles DiD1, D2D2, called base-circles.

• It is evident that the radii of the base-circles bear to each other the same proportions as the radii of the pitch-circles, and also that CiPi=1C1.cos obliquity ~ (27)

• To find the figures of the teeth, draw on a flat surface circular arcs ID1, ID,, with the radii A11, A, I; those arcs will be the developments of arcs of the pitchcircles Bil, B, I, when the conical surfaces AiBf I, A,B~ I are spread out flat.

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

• 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.

• _______) Now c is constant because the ________ - axes are parallel; therefore the sum of the radii of the pitch ~ circles connected in every - position of the belt is to be ~ constant.

• The radii at the middle and end being thus ~ietermined, make the generating curve an arc either of a circle or of a parabola.

• But a limit is put to th~ diminution of the radii of journals and pivots by the conditions 01 durability and of proper lubrication, and also by conditions 01 strength and stiffness.

• Let v be the common velocity of the two pitch-circles, ri, C2, their radii; then the above equation becomes /1 I

• the radii, thicknesses, refractive indices and distances between the lenses, was solved by L.

• 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.

• For a single lens of very small thickness and given power, the aberration depends upon the ratio of the radii r: r', and is a minimum (but never zero) for a certain value of this ratio; it varies inversely with the refractive index (the power of the lens remaining constant).

• The radii, thicknesses and distances are continually altered until the errors of the image become sufficiently small.

• These constants are determined by the data of the system (radii, thicknesses, distances, indices, &c., of the lenses); therefore their dependence on the refractive index, and consequently on the colour, are calculable (the formulae are given in Czapski-Eppenstein, G'rundziige der Theorie der optischen Instrumente (1903, p. 166).

• (b) Two thin lenses in contact: let 4) i and 4) 2 be the powers corresponding to the lenses of refractive indices n i and n 2 and radii r' 1, r" i, and r' 2, r" 2 respectively; let 4) denote the total power, and dc/), 2 the changes of 4), n l, and n 2 with the colour.

• Io); the values of the four radii must satisfy the equations (2) and (4).

• From this he showed that the rise of the liquid in tubes of the same substance is inversely proportional to their radii.

• He 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.

• 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.

• We may also write ur 1 = I +zu 1+ &c., since z is very small compared with u, and expressing u in terms of w by (25), (we find l 21- mv i fi(z) i I +z(c R w + ' R 2 w) do) = 27rmoti(z) I -f-ZZ (Ki + R2/ This then expresses the work done by the attractive forces when a particle m is brought from an infinite distance to the point P at a distance z from a stratum whose surface-density is a, and 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.

• I 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.

• In 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.

• the original position of the moving point on the fixed circle), a and b being the radii of the fixed and rolling circles, and 0 the angle through which the line joining the centres of the two circles has passed.

• 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 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 epicychid when the radii of the circles are equal is the cardioid (q.v), and the corresponding trochoidal curves are limacons.

• an epicycloid in which the radii of the fixed and rolling circles are equal.

• 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.

• 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.

• 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.

• Draw any number of radii of the two circles, and from the points of intersection with the major circle draw lines parallel to the minor axis, and from the points of intersection with the minor circle draw lines parallel to the major axis.

• 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.

• (Reference should be made to the article Crystallography for illustrations, and for applications of these phenomena to the determination of crystal form.) With an uniaxal plate perpendicular to the optic axis, the curves of constant retardation are concentric circles and the lines of like polarization are the radii: thus with polarizer and analyser regulated for extinction, the pattern consists of a series of bright and dark circles interrupted by a black cross with its arms parallel to the planes of polarization and analysation.

• 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.

• If pencils proceed from media of high optical density to media of low density, and have a semi-aperture greater than the critical angle, total reflection occurs; in such cases no plane surface can be employed, hence front lenses have small radii of curvature in order to permit the wide pencils to reach the air (see fig.

• 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.