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.
It is the envelope of circles described on the central radii of an ellipse as diameters.
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 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.
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 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.
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.
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.
Primary, secondary and spurious bows were formed, and their radii measured; a comparison of these observations exhibited agreement with Airy's analytical values.
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.
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 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.
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.
Starting with the stem forms the descendants of which have passed through either persistent or changed habitats, we reach the underlying idea of the branching law of Lamarck or the law of divergence of Darwin, and find it perhaps most clearly expressed in the words "adaptive radiation" (Osborn), which convey the idea of radii in many directions.
Among extinct Tertiary mammals we can actually trace the giving off of these radii in all directions, for taking advantage of every possibility to secure food, to escape enemies and to reproduce kind; further, among such well-known quadrupeds as the horses, rhinoceroses and titanotheres, the modifications involved in these radiations can be clearly traced.
Because of the repetition of analogous physiographic and climatic conditions in regions widely separated both in time and in space, we discover that continental and local adaptive radiations result in the creation of analogous groups of radii among all the vertebrates and invertebrates.
The figure included by two radii and an arc is a " sector," e.g.
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 22° and 46 °, and exhibiting the colours of the spectrum in a confused manner, the only decided tint being the red on the inside.
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.
The remiges and rectrices indicate perfect feathers, with shaft and complete vanes which were so neatly finished that they must have possessed typical radii and hooklets.
It has now been firmly established, both experimentally and mathematically, that coronae are due to diffraction by the minute particles of moisture and dust suspended in the atmosphere, and the radii of the rings depend on the size of the diffracting particles.
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.
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.
The radii, thicknesses, refractive indices and distances between the lenses, was solved by L.
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.
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
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.
A condition equivalent to the above, and necessarily connected with it, is, that at each pair of points of contact the inclinations of the curves to their radii-vectores shall be equal and contrary; or, denoting by r1, rf the radii-vectores at any given pair of points of contact, and s the length of the equal arcs measured from a certain fixed pair of points of contact dri/ds= drm/ds; (18)
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)