curvature

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

139(2) A theorem relating to the apparent curvature of the geocentric path of a comet.

123In general it is not curvature, but change of curvature, that presents difficulty in the laying-out of a line.

126In very many cases the pollen is carried to the stigma by elongation, curvature or some other movement of the filament, the style or stigma, or corolla or some other part of the flower, or by correlated movements of two or more parts.

104This result created a great sensation, and proved that Transatlantic electric wave telegraphy was quite feasible and not inhibited by distance, or by the earth's curvature even over an arc of a great circle 3000 m.

84If 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.

84Leonhard Euler, in his paper on curvature in the Berlin Memoirs for 1760, had considered, not the normals of the surface, but the normals of the plane sections through a particular normal, so that the question of the intersection of successive normals of the surface had never presented itself to him.

61Crystals of blende are of very common occurrence, but owing to twinning and distortion and curvature of the faces, they are often rather complex and difficult to decipher.

61He determined the "elastic curve," which is formed by an elastic plate or rod fixed at one end and bent by a weight applied to the other, and which he showed to be the same as the curvature of an impervious sail filled with a liquid (lintearia).

63Moreover the greater depths of the curves (or "curvature powers") in itself neutralize more or less the advantages obtained from the reduced irrationality of dispersion.

68If a surface intended to be flat is affected with a slight general curvature, a remedy may be found in an alteration of focus, and the remedy is the less complete as the reflection is more oblique.

611In the long range high angle fire the shot ascends to such a height that the correction for the tenuity of the air becomes important, and the curvature 4)-8 of an arc should be so chosen that 4)y 0, the height ascended, should be limited to about moo ft., equivalent to a fall of I inch in the barometer or 3% diminution in the tenuity factor T.

53He suggests that the propagation of earthquake disturbances is probably affected by the curvature of the surface of the globe, which may act like a whispering gallery.

54r, Radius of curvature, formula (1).

54The object, however, can be fully attained only if the scale of the map is sufficiently large, if the horizontal and vertical scales are identical, so that there shall be no exaggeration of the heights, and if regard is had, eventually, to the curvature of the earth's surface.

58In the shape and curvature of the horns, which at first incline outwards and forwards, and then bend somewhat upwards and inwards, this breed of cattle resembles the aurochs and the (by comparison) dwarfed park-breeds.

44The face has the ordinary gazelle-markings; but the rather short horns - which are wanting in the female - have a peculiar upward and forward curvature, unlike that obtaining in the gazelles FIG.

44The semi-elliptical shape of the arches, the variation of span, the _ slight curvature of the 26:0'=-----.

44Then 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.

44Divide 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.

44arabica, in which the horns have a somewhat S-shaped curvature in profile.

4721 (d); in this case a convex mirror of different curvature is employed, the equivalent focus of the combination being 80 ft.

32Apollonius' genius takes its highest flight in Book v., where he treats of normals as minimum and maximum straight lines drawn from given points to the curve (independently of tangent properties), discusses how many normals can be drawn from particular points, finds their feet by construction, and gives propositions determining the centre of curvature at any point and leading at once to the Cartesian equation of the evolute of any conic.

33Proposition 30 describes the construction of a curve of double curvature called by Pappus the helix on a sphere; it is described by a point moving uniformly along the arc of a great circle, which itself turns about its diameter uniformly, the point describing a quadrant and the great circle a complete revolution in the same time.

33He was well aware of the failures of all attempts to perfect telescopes by employing lenses of various forms of curvature, and accordingly proposed the form of reflecting telescope which bears his name.

33Since the curvature powers of the positive lenses are equal, the partial dispersions of the two glasses may be simply added together, and we then have: [0.543 +0.3741 The proportions given on the lower line may now be compared with the corresponding proportional dispersions for borosilicate flint glass 0.658, closely resembling the type 0.164 of Schott's list, viz.: [0.658 (A D = I.546) 50' 11 A slight increase in the relative power of the first lens of 0.543 would bring about a still closer correspondence in the rationality, but with the curves required to produce an object-glass of this type of 6 in.

34Starting from an observation of Marconi's, a number of interesting facts have been accumulated on the absorbing effect of sunlight on the propagation of long Hertzian waves through space, and on the disturbing effects of atmospheric electricity as well as upon the influence of earth curvature and obstacles of various kinds interposed in the line between the sending and transmitting stations.4 Electric wave telegraphy has revolutionized our means of communication from place to place on the surface of the earth, making it possible to communicate instantly and certainly between places separated by several thousand miles, whilst The Electrician, 1904, 5 2, p. 407, or German Pat.

35- This section of the Atlas, known to the inhabitants of Morocco by its Berber name, Idraren Draren or the " Mountains of Mountains," consists of five distinct ranges, varying in length and height, but disposed more or less parallel to one another in a general direction from south-west to north-east, with a slight curvature towards the Sahara.

36A linear error in the spacing, and a general curvature of the lines, are eliminated in the ordinary use of a grating.

212Much discussion has taken place as to the utility of such curvature.

10In contrast, female competitors will cross one leg in front of the other to accentuate the curvature of their bodies.

10The horns of the bucks are heavy, and have a peculiar forward curvature at the tips; the colour of the coat is red-fawn, with a broad brown band down the back.

11de Paris, 1781), which, while giving a remarkably elegant investigation in regard to the problem 3f earth-work referred to in the title, establishes in connexion with it his capital discovery of the curves of curvature of a surface.

11The radius of curvature of a catenary is equal and opposite to the portion of the normal intercepted by the directrix of the catenary.

11Hence it is clear that if the two positive lenses of equal curvature power of o 60 and 0.102 respectively are combined with a negative lens of light flint o 569, then a triple objective, having no secondary spectrum (at any rate with respect to the blue rays), may be obtained.

12Monge's memoir just referred to gives the ordinary differential equation of the curves of curvature, and establishes the general theory in a very satisfactory manner; but the application to the interesting particular case of the ellipsoid was first made by him in a later paper in 1795.

13where p is the radius of curvature of the path at P, the tangential and normal accelerations are also expressed by v dv/ds and v1/p, respectively.

13When a root comes in contact at its tip with scme hard body, such as might impede its progress, a curvature of the growing part is set up, which takes the young tip away from the stone, or what-not, with which it is in contact.

14Let it be supposed that two positive lenses of equal curvature powers are made out of these two glasses, then in order to represent the combined dispersion of the two together the two 0µ's for each spectral region may be added together to form 0'µ as in the line below, and then, on again expressing the partial z'µ in terms of L'µ (C to F) we get the new figures in the bottom row beneath the asterisks.

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

00In mathematics, the "caustic surfaces" of a given surface are the envelopes of the normals to the surface, or the loci of its centres of principal curvature.

00The vertebral column of the gorilla differs from that of man in its curvature and other characters, as also does the conformation of its narrow pelvis.

00In narrow rock gorges extremely interesting and complex problems relating to the combined action of horizontal and vertical stresses arise, and in some such cases it is evident that much may be done by means of horizontal curvature to reduce the quantity of masonry without reduction of strength.

00and five of them have cracked vertically, owing apparently to the impossibility of the base of the dam partaking of the changes of curvature induced by changes of temperature and of moisture in the upper parts.

00The Jurua is the next great southern affluent of the Amazon west of the Purus, sharing with this the bottom of the immense inland Amazon depression, and having all the characteristics of the Purus as regards curvature, sluggishness and general features of the low, half-flooded forest country it traverses.

00Many well-known derivative curves present themselves in this manner; thus the variable curve may be the normal (or line at right angles to the tangent) at any point of the given curve; the intersection of the consecutive normals is the centre of curvature; and we have the evolute as at once the locus of the centre of curvature and the envelope of the normal.

00- In conclusion a little may be said as to curves of double curvature, otherwise twisted curves or curves in space.

00On the other hand, the great tawny draught cattle of Spain seem to indicate mixture with a different stock, the horns having a double curvature, quite different from the simple one of the aurochs type.

00The embryo has now lost its spiral curvature, and becomes completely doubled upon itself, the hind end being in contact with the mouth (fig.

00When the pupil regulates the aperture of the rays producing the image the aberrations of the ordinary lenses increase considerably with the magnification, or, what amounts to the same thing, with the increase in the curvature of the surfaces.

00Westien made use of two Chevalier-Briicke's simple microscopes with their long working distances in order to form an instrument in which the curvature of the image was not entirely avoided.

00The usual concept of risk aversion generated through utility curvature seems inappropriate for modeling loss aversion.

00I've not figured out the curvature numbers that will give flat panels (hard chines ).

00AtmDCC An atmospheric dispersion compensator and field corrector (AtmDCC) is required to correct for atmospheric refraction and field curvature.

00In all such cases, there is no component to plan convexity, any curvature being entirely orthogonal to the xy plane.

00By measuring the curvature of the tracks, the energy of the electrons to be estimated.

00I like the way the plants follow the curvature of the vase.

00What causes curvature of the spine in children and how can surgery be used to minimize its effect?

00Flatten the lumbar curvature by raising the patient's knees.

00Sleeping on a firm bed with no more than one pillow can help prevent spinal curvature.

00Seed surface patches are formed by grouping neighboring pixels whose mean and Gaussian curvature have the same sign.

00Abstract: I will spend half the talk motivating the search for constant scalar curvature Kahler metrics.

00Spinal curvature Scoliosis is a lateral curvature of the spine, often with a rotational element.

00curvature scale.

00curvature scale space filtering of a 2-D contour generally produces successively smoother versions of that contour.

00Using the surface curvature of points along the silhouette a new silhouette can be predicted by applying a certain transformation.

00Fast visible image of an ELM in MAST Measurements of neoclassical island evolution appearing to confirm the strong stabilizing role of field curvature effects.

00Linearised horizontally to remove the effects of the earth's curvature.

00curvature of spacetime is maximal.

00curvature of space-time caused by the motions of matter.

00curvature of the spine.

00curvature of the earth of every side.

00curvature of the universe.

00curvature of a surface were to be continued, this would result in a sphere.

00glistening lakes and marshes within, and see the curvature of the earth on the horizon.

00inexact graph matching, surface reconstruction and surface curvature analysis.

00plano-convex lens is placed with its curved surface with radius of curvature R resting on a plane glass surface.

00radius of curvature of 30 cm.

00scalar curvature Kahler metrics.

00Symptoms of idiopathic scoliosis In children, scoliosis In children, scoliosis usually has no symptoms at all until the curvature becomes severe.

0014); and that in y 3 would be caused by a curvature such that there is a point of inflection at the middle of each line (fig.

00(22); and for the curvature, Cornu remarks that this equation suffices to determine the general character of the curve.

00will move from G to G 1 the reduced distance G1G2 = c (P/W); and if B, called the centre of buoyancy, moves to B 1, along the curve of buoyancy BB 1, the normal of this curve at B 1 will be the new vertical B1G1, meeting the old vertical in a point M, the centre of curvature of BB I, called the metacentre.

00tons, (4) w denoting the density of water in tons/ft.', and W =wV, for a displacement of V ft.3 This couple, combined with the original buoyancy W through B, is equivalent to the new buoyancy through B, so that W.BB 1 =wAk 2 tan 8, (5) BM =BB 1 cot B=Ak e /V, (6) giving the radius of curvature BM of the curve of buoyancy B, in terms of the displacement V, and Ak e the moment of inertia of the water-line area about an axis through F, perpendicular to the plane of displacement.

00(12) Along the stream line xBAPJ, t ' =0, u=ae-" c bl, n; (13) and over the jet surface JPA, where the skin velocity is Q, - q = - Q, u = ae rs Q /m = ae rs lc, (14) ds denoting the arc AP by s, starting at u = a; a ' ch nS2=cos nB= -a' u u - - a b' (15) a l a - b l u - a' a-a' u-b' co > u = ae'" S " c > a, and this gives the intrinsic equation of the jet, and of curvature ds '&1) _ i dw i dw dS2 P= - dO = Q a0 - Q as2 = Q c u-b d (u -a.u -a') _ ?

00In these cases the curvature of the trajectory becomes considerable, and the formulae employed in direct fire must be modified; the method generally employed is due to Colonel Siacci of the Italian artillery.

00But, as originally pointed out by Euler, the difficulty can be turned if we notice that in the ordinary trajectory of practice the quantities i, cos i, and sec i vary so slowly that they may be replaced by their mean values,, t, cos 7 7, and sec r t, especially if the trajectory, when considerable, is divided up in the calculation into arcs of small curvature, the curvature of an arc being defined as the angle between the tangents or normals at the ends of the arc.

00In the application of Siacci's method to the calculation of a trajectory in high angle fire by successive arcs of small curvature, starting at the beginning of an arc at an angle 4) with velocity v4), the curvature of the arc 4-8 is first settled upon, and now (80) n=1(0+0) is a good first approximation for n.

00(See River -HoG.) The recently discovered Hylochoerus of the equatorial forestdistricts of Africa comes nearest to the under-mentioned warthogs, but the skull is of a much less specialized type, while the upper tusks are much smaller although they have the same general curvature and direction, and the cheek-teeth lack the peculiar characteristics of those of Phacochoerus, although they present a certain approximation thereto.

00The under surface of the left lobe is concave for the interior surface of the stomach (see Alimentary Canal: Stomach Chamber), while a convexity, known as the tuber omentale, fits into the lesser curvature of that organ.

00duced by euclidian methods from the definition include the following: the tangent at any point bisects the angle between the focal distance and the perpendicular on the directrix and is equally inclined to the focal distance and the axis; tangents at the extremities of a focal chord intersect at right angles on the directrix, and as a corollary we have that the locus of the intersection of tangents at right angles is the directrix; the circumcircle of a triangle circumscribing a parabola passes through the focus; the subtangent is equal to twice the abscissa of the point of contact; the subnormal is constant and equals the semilatus rectum; and the radius of curvature at a point P is 2 (FP) 4 /a 2 where a is the semilatus rectum and FP the focal distance of P.

00By directing the telescope to a distant object, or to the intersection of the webs of a fixed collimating telescope (see Transit Circle), it is easy to measure the effect of a small change of zenith distance of the axis of the telescope in terms both of the level and of the micrometer screw, and thus, if the levels are perfectly sensitive and uniform in curvature and graduation, to determine the value of one division of each level in terms of the micrometer screw.

00The whole of the foregoing reasonings are applicable, not merely when acm and bbb are actual cylinders, but also when they are the osculating cylinders of a pair of cylindroidal surfaces of varying curvature, A and B being the axes of curvature of the parts of those surfaces which are in contact for the instant under consideration.

00The same holds for the errors depending upon the angle of the field of view, w: astigmatism, curvature of field and distortion are eliminated for a definite value, w*; " zones of astigmatism, curvature of field and distortion " attend smaller values of w.

00(2) Largest field of view; necessary corrections are - for astigmatism, curvature of field and distortion; errors of the aperture only slightly regarded; examples - photographic widest angle objectives and oculars.

00Photog., 1891, 5, p. 225; 18 93, 7, p. 221), cemented objectives of thin lenses permit the elimination of spherical aberration on the axis, if, as above, the collective lens has a smaller refractive index; on the other hand, they permit the elimination of astigmatism and curvature of the field, if the collective lens has a greater refractive index (this follows from the Petzval equation; see L.

00Then if 0 is the centre of curvature in the plane of the paper, and BO =u, I _ cos sinew u R 1 R2 Let POQ=o, PO=r, PQ=f, BP=z, f 2 = u 2 +r 2 -2ur cos 0 (26) The element of the stratum at Q may be expressed by ou t sin o do dw, or expressing do in terms of df by (26), our 1fdfdw.

00The 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.

00The 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.

00If 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.

00A shaving or make-up mirror of this type has a radius of curvature of 30 cm.

00Symptoms of idiopathic scoliosis In children, scoliosis usually has no symptoms at all until the curvature becomes severe.

00Here, after demonstrating that shear deformation alone makes little qualitative difference, the case of initial curvature is examined in some detail.

00If the specular surface is of high curvature the light will be tend to be focused.

00People with curvature of the spine may need a chair with additional support, while others may find that a moderate support is best.

00Square If you're male, you can still get by with squarish lenses if you'd like, though an aviator may make a better choice due to the slight curvature of the style.

00ARC Prismatic polycarbonate lenses: ARC stands for "accurate radius curvature," which means that the lens is thickest at the optical center, then thins toward the edges.

00Curvature of the spin (scoliosis) may be present, elevated blood pressure, and abnormalities in height, weight, and head size may also be noticed on physical examination.

00Also, infants with this type of EDS have an abnormal curvature of the spine (scoliosis).

00Kyphoscoliosis-Abnormal front-to-back and side-to-side curvature of the spine.

00Scoliosis-An abnormal, side-to-side curvature of the spine.

00The child's face may also be long and narrow, and he or she may have a noticeable curvature of the spine.

00Scoliosis. Scoliosis, or curvature of the spine, is a disorder in which the vertebrae that make up the spine twist out of line from side to side into an S-shape or a spiral.

00Kyphosis. Kyphosis is an abnormal outward curvature of the spine at the back, sometimes called hunchback when it occurs in the upper back.

00The doctor simply asks the child to bend forward while the back is examined for changes in the curvature.

00Curves of 40 degrees or more are highly likely to worsen, even in an adult, because the spine is so badly imbalanced that the force of gravity will increase the curvature.

00If the spinal curvature increases to 40 or 50 degrees, the child may require surgery in order to prevent lung problems, back pain, and further deformity.

00Kyphosis-An extreme, abnormal outward curvature of the spine, with a hump at the upper back.

00They may be very far-sighted or near-sighted and may have other defects in the curvature of the lens of the eye (astigmatism) that cause images to appear unfocused.

00A side-to-side curvature of the spine (scoliosis) occurs in many cases, and may become severe.

00Spastic, hypertonic muscles can cause serious orthopedic problems, including curvature of the spine (scoliosis), hip dislocation, or contractures.

00Scoliosis is a side-to-side curvature of the spine.

00While a small degree of lateral curvature does not cause any medical problems, larger curves can cause postural imbalance and lead to muscle fatigue and pain.

00Infantile: Curvature appears before age three.

00Juvenile: Curvature appears between ages three and ten.

00Adolescent: Curvature usually appears between ages of ten and 13, near the beginning of puberty.

00Adult: Curvature begins after physical maturation is completed.

00An x ray is also used to document spinal maturity, any pelvic tilt or hip asymmetry, and the location, extent, and degree of curvature.

00These angles are referred to when the angle of the curvature is discussed.

00Treatment decisions for scoliosis are based on the degree of curvature, the likelihood of significant progression, and the presence of pain, if any.

00Bracing cannot correct curvature but may be effective in halting or slowing progression.

00The surgical procedure for scoliosis is called spinal fusion, because the goal is to straighten the spine as much as possible and then to fuse the vertebrae together to prevent further curvature.

00Fusion of the spine makes it rigid and resistant to further curvature.

00Some children develop a curvature in the spine, flat feet, and a heart condition known as mitral valve prolapse.

00Tendency toward scoliosis (a curvature of the spine) is present.

00Scoliosis (a curvature of the spine) is present.

00Scoliosis (curvature of the spine) is likely.

00Weakening of the trunk muscles around this age often leads to scoliosis (a side-to-side spine curvature) and kyphosis (a front-to-back curvature).

00Surgery is recommended at a much lower degree of curvature for DMD than for scoliosis due to other conditions, since the decline in respiratory function in DMD makes surgery at a later time dangerous.

00Problems with muscle tone and nervous system abnormalities will affect the development of motor skills, possibly resulting in scoliosis (curvature of the spine) and esotropia (crossed eyes).

00Chordee-An abnormal curvature of the penis.

00It is present at birth, and children exhibit severe contractures of the joints, resulting in limb deformity; spinal curvature; deformities of the chest wall; difficulties breathing; abnormally small jaw; and upper eyelid droop (ptosis).

00Surgery may be necessary for spinal curvature and severe contractures.

00Rarely is spine curvature so pronounced in Scheuermann disease that the individual needs to wear a brace or have surgical intervention.

00The back is stacked up so that the curvature of the head is more visible and the hair nearest the nape of the neck is short and unmoving.

00They can pick the curvature of the wings, the pattern style, colors, and select how they wish the antennas to look.

00The program is designed to help relieve back pain and other symptoms of scoliosis, a condition caused by an S- or C-shaped sideways curvature of the spine.

00Spine: Focus on lengthening the spine to help reduce the characteristic curvature of scoliosis.

00He blamed corsets for tuberculosis, cancer and curvature of the spine and many other illnesses which in fact were due to the prevailing sanitary conditions.

00Can prevent your shoulders from drooping, poor posture, backaches and spinal curvature.

00The condition impacts bone and cartilage development and has many physical symptoms beyond short stature including club foot, hitchhiker's thumb and curvature of the spine.

00However, many women prefer a curved white area that follows the natural curvature of the nail bed.

00The smoothest and safest running is, in fact, attained when a " transition," " easement " or " adjustment " curve is inserted between the tangent and the point of circular curvature.

01Under the general heading "Geometry" occur the subheadings "Foundations," with the topics principles of geometry, non-Euclidean geometries, hyperspace, methods of analytical geometry; "Elementary Geometry," with the topics planimetry, stereometry, trigonometry, descriptive geometry; "Geometry of Conics and Quadrics," with the implied topics; "Algebraic Curves and Surfaces of Degree higher than the Second," with the implied topics; "Transformations and General Methods for Algebraic Configurations," with the topics collineation, duality, transformations, correspondence, groups of points on algebraic curves and surfaces, genus of curves and surfaces, enumerative geometry, connexes, complexes, congruences, higher elements in space, algebraic configurations in hyperspace; "Infinitesimal Geometry: applications of Differential and Integral Calculus to Geometry," with the topics kinematic geometry, curvature, rectification and quadrature, special transcendental curves and surfaces; "Differential Geometry: applications of Differential Equations to Geometry," with the topics curves on surfaces, minimal surfaces, surfaces determined by differential properties, conformal and other representation of surfaces on others, deformation of surfaces, orthogonal and isothermic surfaces.

01When the curve after its steep descent has almost reached the axis, it bends aside sharply and becomes a nearly horizontal straight line; the authors suggest that the critical temperature should be defined as that corresponding to the point of maximum curvature.

01The former measures the thickness of the primary focal line, and the latter measures its curvature.

01The 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 '.

01The spectrum will be straight if the twoprismsaresimilar in dispersive property, but if one of them is con structed of a material which possesses any peculiarity in this respect it will be revealed by the curvature of the spectrum.

01The general colour is bluegrey with black "points" and white markings and belly; and the horns of the rams are olive-brown and nearly smooth, with a characteristic backward curvature.

01Until Newton's discovery of the different refrangibility of light of different colours, it was generally supposed that object-glasses of telescopes were subject to no other errors than those which arose from the spherical figure of their surfaces, and the efforts of opticians were chiefly directed to the construction of lenses of other forms of curvature.

01where 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.

01P we have (T + T) sin ai,L, or T4~, or Ts/p, where p is the radius of curvature.

01Suppose, for example, that we have a light string stretched over a smooth curve; and let Rs denote the normal pressure (outwards from the centre of curvature) on bs.

01The tension is constant, and the pressure per unit length varies as the curvature.

01the curve must be a geodesic, and that the normal pressure per unit length must vary as the principal curvature of the curve.

01The 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.

01Nearly Epicycloidal Teeth: Williss Method.To facilitate the drawing of epicycloidal teeth in practice, Willis showed how to approximate to their figure by means of two circular arcsone concave, for the flank, and the other convex, for the faceand each having for its radius the mean radius of curvature of the epicycloidal arc. \Villiss formulae are founded on the following properties of epicycloids Let R be the radius of the pitch-circle; r that of the describing circle; 8 the angle made by the normal TI to the epicycloid at a given point T, with a tangent-to the circle at Ithat is, the obliquity of the action at T.

01Then the radius of curvature of the epicycloid at T is For an internal epicycloid, p =4r sin o~1

01Also, to find the position of the centres of curvature relatively to the pitch-circle, we have, denoting the chord of the describing circle TI bye, c=2r sin 0; and therefore For the flank, p C=2r sine ~~_r} (29) For the face, pC2r sin 8 ~-~

01By 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.

01He 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.

01If the above errors be eliminated, the two astigmatic surfaces united, and a sharp image obtained with a wide aperture - there remains the necessity to correct the curvature of the image surface, especially when the image is to be received upon a plane surface, e.g.

01From this it follows that correctness of drawing depends solely upon the principal rays; and is independent of the sharpness or curvature of the image field.

01Trans., 1830, 3, p. 1) is fulfilled in all systems which are symmetrical with respect to their diaphragm (briefly named " symmetrical or holosymmetrical objectives "), or which consist of two like, but different-sized, components, placed from the diaphragm in the ratio of their size, and presenting the same curvature to it (hemisymmetrical objectives); in these systems tan w'/ tan w= 1.

01The aberrations of the third order are: (1) aberration of the axis point; (2) aberration of points whose distance from the Aberra- axis is very small, less than of the third order - the tions of deviation from the sine condition and coma here fall together in one class; (3) astigmatism; (4) curvature of the field; (5) distortion.

01In attempting to calculate the effect of this surface-tension in determining the form of a drop of the liquid, Segner took account of the curvature of a meridian section of the drop, but neglected the effect of the curvature in a plane at right angles to this section.

01des Sciences, 1787, p. 506) asserted that " by supposing the adherence of the particles of a fluid to have a sensible effect only at the surface itself and in the direction of the surface it would be easy to determine the curvature of the surfaces of fluids in the neighbourhood of the solid boundaries which contain them; that these surfaces would be linteariae of which the tension, constant in all directions, would be everywhere equal to the adherence of two particles, and the phenomena of capillary tubes would then present nothing which could not be determined by analysis."

01He thus showed that at a curved part of the surface, a superficial particle would be urged towards the centre of curvature of the surface, and he gave reasons for concluding that this force is proportional to the sum of the curvatures of the surface in two normal planes at right angles to each other.

01He 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.

01Thomson (afterwards Lord Kelvin) investigated the effect of the curvature of the surface of a liquid on the thermal equilibrium between the liquid and the vapour in contact with it.

01Since 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.

01If 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,.

01Let 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.

01section whose radius of curvature is R1.

01Integrating 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.

01We 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.

01It is also practically independent of the curvature of the surface, although it appears from the mathematical theory that there is a slight increase of tension where the mean curvature of the surface is concave, and a slight diminution where it is convex.

01When 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.

01Suppose 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.

01(io) Now ds - sin a (II) The radius of curvature of the meridian section is ds R1= a.

01(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).

01This 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.

01I 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.

01- When the internal pressure is equal to the external, the film forms a surface of which the mean curvature at every point is zero.

01We 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.

01Hence 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.

01The mean curvature of these surfaces is therefore convex towards the axis.

01The mean curvature of these surfaces is, therefore, concave towards the axis.

01Now if the pressure is equal on both sides of a liquid film, and if its mean curvature is zero, it will be in equilibrium.

01If the mean curvature is convex towards the axis the film will move from the axis.

01If the mean curvature is concave towards the axis the film will tend to approach the axis.

01Now if the displacement z be everywhere very small, the curvature in the planes parallel to xz and yz will be d 2 z/dx 2 and d 2 z/dy e respectively, and if T is the surface-tension the whole upward force will be d 2 z d2zl T (4x 2 + + (p - o) gz.

01The wing of the bird, like that of the insect, is concavo-convex, and more or less twisted upon itself when extended, so that the anterior or thick margin of the pinion presents a different degree of curvature to that of the posterior or thin margin.

01The 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.

01The marked curvature of the vertebral column, by breaking the shock to the neck and head in running and leaping, likewise favours the erect position.

01In New South Wales thirteen thin concrete dams, dependent upon horizontal curvature for their resistance to water pressure, have been constructed in narrow gorges at comparatively small cost to impound water for the use of villages.

01In geology, a fold is a bend or curvature in the stratified rocks of the earth's crust, whereby they have been made to take up less horizontal space.

01In every instance therefore where, in walking over the surface, we traverse a series of strata which gradually, and without dislocations, increase or diminish in inclination, we cross part of a great curvature in the strata of the earth's crust.

01The curvature occasionally shows itself among horizontal or gently inclined strata in the form of an abrupt inclination, and then an immediate resumption of the previous flat or sloping character.

01A further aberration which can only be overcome with difficulty, and even then only partially, is the " curvature of the field, " i.e.

0146) is made entirely of glass, and is in the form of a paraboloid, having on the top a spherical hole, of such a curvature that all entering rays, r r' r", parallel to the axis, after their reflection on the surface of the paraboloid, traverse the spherical surface perpendicularly and unite in F, the centre of the sphere.

01Owing to the curvature of the image, all parts of the object are not seen distinctly at one and the same time.

01The curvature of the range around the Brazilian massif, and the position of the zone of older rocks upon the eastern flank, led Suess to the conclusion that the Andes owe their origin to an overthrust from east to west, and that the Vorland lies beneath the Pacific. In the south Wehrli and Burckhardt maintain that the thrust came from the west, and they look upon the ancient rocks of Argentina as the Vorland.

01There is also a very slight curvature, but this is to be expected in a turned ebony shank of this age.

01shear deformation alone makes little qualitative difference, the case of initial curvature is examined in some detail.

01specular surface is of high curvature the light will be tend to be focused.

01This destroys the beauty of the field equations, which attribute the source of curvature entirely to matter as represented by the stress-energy tensor.

013, d.mes.) and ventral mesenteries, the latter following the outer curvature of the loop of the alimentary canal.

01Mendel Dessau was a poor scribe - a writer of scrolls - and his son Moses in his boyhood developed curvature of the spine.

01(On the continent of Europe, however, six-wheeled vehicles are to be found much longer than those employed in Great Britain.) This difficulty is avoided by providing the vehicles with four axles (or six in the case of the largest and heaviest), mounted in pairs (or threes) at each end in a bogie or swivel truck, which being pivoted can move relatively to the body and adapt itself to the curvature of the line.

01The adaption of these gores to the curvature of the sphere calls for great care.

01In like manner, the term in y 2 corresponds to a general curvature of the lines (fig.

01The term in y corresponds to a variation of curvature in crossing the grating (fig.

01In Great Britain the curvature is defined by stating the length of the radius, expressed in chains (i chain=66 ft.), in America by stating the angle subtended by a chord ioo ft.

02The amount of superelevation required to prevent derailment at a curve can be calculated under perfect running conditions, given the radius of curvature, the weight of the vehicle, the height of the centre of gravity, the distance between the rails, and the speed; but great experience 1 See The Times Engineering Supplement (August 22, 1906), p. 265.

02Curvature of the primary focal line having a very injurious effect upon definition, it may be inferred from the excellent performance of these gratings that y is in fact small.

02The 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.'

02if r denotes the radius of curvature of the stream line, so that I dp + dV - dH _ dq 2 q2 (6) p dv dv dv dv - r ' the normal acceleration.

02Along the path of a particle, defined by the of (3), _ c) sine 2e, - x 2 + y2 = y a 2 ' (Io) sin B' de' _ 2y-c dy 2 ds ds' on the radius of curvature is 4a 2 /(ylc), which shows that the curve is an Elastica or Lintearia.

02

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