# Elliptic Sentence Examples

He was also the author of important papers in which he extended to complex quadratic forms many of Gauss's investigations relating to real quadratic forms. After 1864 he devoted himself chiefly to

**elliptic**functions, and numerous papers on this subject were published by him in the Proc. Lond.Under the general heading "Analysis" occur the subheadings "Foundations of Analysis," with the topics theory of functions of real variables, series and other infinite processes, principles and elements of the differential and of the integral calculus, definite integrals, and calculus of variations; "Theory of Functions of Complex Variables," with the topics functions of one variable and of several variables; "Algebraic Functions and their Integrals," with the topics algebraic functions of one and of several variables,

**elliptic**functions and single theta functions, Abelian integrals; "Other Special Functions," with the topics Euler's, Legendre's, Bessel's and automorphic functions; "Differential Equations," with the topics existence theorems, methods of solution, general theory; "Differential Forms and Differential Invariants," with the topics differential forms, including Pfaffians, transformation of differential forms, including tangential (or contact) transformations, differential invariants; "Analytical Methods connected with Physical Subjects," with the topics harmonic analysis, Fourier's series, the differential equations of applied mathematics, Dirichlet's problem; "Difference Equations and Functional Equations," with the topics recurring series, solution of equations of finite differences and functional equations.All the major planets and many of the minor planets revolve in

**elliptic**g FIG.For the liquid filling the interior of a rotating

**elliptic**cylinder of cross section x2/a2+y2/b2 = 1, /(4) = m l (x 2 / a2 - - y2 /b 2) (5) with V21G1' =-2R =-2 m 1 (I / a2 + 21b2), 214 = m l (x2 / a2 + y2 / b2) - IR(x2+y2) = I R (x2 - y2) (a 2 - b2)/(a2+b2), cp 1 = Rxy (a 2 - b2)/(a2 +b2), w1 = cb1 +% Pli = - IiR(x +yi)2(a2b2)/(a2+b2).The extension to the case where the liquid is bounded externally by a fixed ellipsoid X= X is made in a similar manner, by putting 4 = x y (x+ 11), (io) and the ratio of the effective angular inertia in (9) is changed to 2 (B0-A0) (B 1A1) +.a12 - a 2 +b 2 a b1c1 a -b -b12 abc (Bo-Ao)+(B1-A1) a 2 + b 2 a1 2 + b1 2 alblcl Make c= CO for confocal

**elliptic**cylinders; and then _, 2 A? ?From Berlin he passed to Freiberg, and here he made his brilliant researches in the theory of functions,

**elliptic**, hyperelliptic and a new class known as Abelians being particularly studied.The above expressions for the capacity of an ellipsoid of three unequal axes are in general

**elliptic**integrals, but they can be evaluated for the reduced cases when the ellipsoid is one of revolution, and hence in the limit either takes the form of a long rod or of a circular disk.They are fleshy shrubs, with rounded, woody stems, and numerous succulent branches, composed in most of the species of separate joints or parts, which are much compressed, often

**elliptic**or suborbicular, dotted over in spiral lines with small, fleshy, caducous leaves, in the axils of which are placed the areoles or tufts of barbed or hooked spines of two forms. The flowers are mostly yellow or reddish-yellow, and are succeeded by pear-shaped or egg-shaped fruits, having a broad scar at the top, furnished on their soft, fleshy rind with tufts of small spines.But Landen's capital discovery is that of the theorem known by his name (obtained in its complete form in the memoir of 1775, and reproduced in the first volume of the Mathematical Memoirs) for the expression of the arc of an hyperbola in terms of two

**elliptic**arcs.His researches on

**elliptic**functions are of considerable elegance, but their great merit lies in the stimulating effect which they had on later mathematicians.AdvertisementOn his return he removed to Berlin, where he lived as a royal pensioner till his death, which occurred on the 18th of February 18 His investigations in

**elliptic**functions, the theory of which he established upon quite a new basis, and more particularly his development of the theta-function, as given in his great treatise Fundamenta nova theoriae functionum ellipticarum (Konigsberg, 1829), and in later papers in Crelle's Journal, constitute his grandest analytical discoveries.Noteworthy are the

**elliptic**form of the chief temples.His first published writings upon the subject consist of two papers in the Memoires de l'Academie Francaise for 1786 upon

**elliptic**arcs.In 1792 he presented to the Academy a memoir on

**elliptic**transcendents.The third volume (1816) contains the very elaborate and now well-known tables of the

**elliptic**integrals which were calculated by Legendre himself, with an account of the mode of their construction.AdvertisementLegendre had pursued the subject which would now be called

**elliptic**integrals alone from 1786 to 1827, the results of his labours having been almost entirely neglected by his contemporaries, but his work had scarcely appeared in 1827 when the discoveries which were independently made by the two young and as yet unknown mathematicians Abel and Jacobi placed the subject on a new basis, and revolutionized it completely.The Exercices de calcul integral consist of three volumes, a great portion of the first and the whole of the third being devoted to

**elliptic**functions.It will thus be seen that Legendre's works have placed him in the very foremost rank in the widely distinct subjects of

**elliptic**functions, theory of numbers, attractions, and geodesy, and have given him a conspicuous position in connexion with the integral calculus and other branches of mathematics.The area is (b 2 +a 2 /2)7r, and the length is expressible as an

**elliptic**integral.They were invented by Gauss to facilitate the computation of

**elliptic**integrals.AdvertisementSome years later he succeeded in showing that Kepler's

**elliptic**orbit for planetary motion agreed with the assumed law of attraction; he also completed the co-ordination with terrestrial gravity by his investigation of the attractions of homogeneous spherical bodies.There are traces of an altar near the Heraeum which was probably older than the great altar of Zeus; this was probably the original centre of worship. The great altar of Zeus was of

**elliptic**form, the length of the lozenge being directed from south-south-west to north-north-east, in such a manner that the axis would pass through the Cronion.It is of conoidal form, with an irregular

**elliptic**base, and rises abruptly to a height of 1114 ft.In Short's first telescopes the specula were of glass, as suggested by Gregory, but he afterwards used metallic specula only, and succeeded in giving to them true parabolic and

**elliptic**figures.The practical difficulty of constructing Gregorian telescopes of good defining quality is very considerable, because if spherical mirrors are employed their aberrations tend to increase each other, and it is extremely difficult to give a true

**elliptic**figure to the necessarily deep concavity of the small speculum.AdvertisementIn

**elliptic**harmonic motion the velocity of P is parallel and proportional to the semi-diameter CD which is conjugate to the radius CP; the hodograph is therefore an ellipse similar to the actual orbit.This applies to an

**elliptic**or hyperbolic orbit; the case of the parabolic orbit may be examired separately or treated as a limiting case.To complete the solution of (2) a third integral is required; this involves in general the use of

**elliptic**functions.The

**elliptic**functions degenerate into simpler forms when ki=o or = I.The n formulae of this type represent a normal mode of free vibration; the individual particles revolve as a rule in

**elliptic**orbits which gradually contract according to the law indicated by the exponential factor.He was much interested, too, in universal algebra, non-Euclidean geometry and

**elliptic**functions, his papers "Preliminary Sketch of Bi-quaternions" (1873) and "On the Canonical Form and Dissection of a Riemann's Surface" (1877) ranking as classics.Among his most remarkable works may be mentioned his ten memoirs on quantics, commenced in 1854 and completed in 1878; his creation of the theory of matrices; his researches on the theory of groups; his memoir on abstract geometry, a subject which he created; his introduction into geometry of the "absolute"; his researches on the higher singularities of curves and surfaces; the classification of cubic curves; additions to the theories of rational transformation and correspondence; the theory of the twenty-seven lines that lie on a cubic surface; the theory of

**elliptic**functions; the attraction of ellipsoids; the British Association Reports, 1857 and 1862, on recent progress in general and special theoretical dynamics, and on the secular acceleration of the moon's mean motion.In the solution the value of an

**elliptic**function is found by means of the arithmetico-geometrical mean.Observe that the radical, square root of a quartic function, is connected with the theory of

**elliptic**functions, and the radical, square root of a sextic function, with that of the first kind of Abelian functions, but that the next kind of Abelian functions does not depend on the radical, square root of an octic function.It is a form of the theorem for the case D = r, that the coordinates x, y, z of a point of the bicursal curve, or in particular the co-ordinates of a point of the cubic, can be expressed as proportional to rational and integral functions of the

**elliptic**functions snu, cnu, dnu; in fact, taking the radical to be r -0 2 .r - k 2 0 2, and writing 8 =snu, the radical becomes = cnu, dnu; and we have expressions of the form in question.A body can move round the sun in an

**elliptic**orbit having the sun in its focus, and describing equal areas in equal times, only under the influence of a force directed towards the sun, and varying inversely as the square of the distance from it.When one or more other bodies form a part of the system, their action produces deviations from the

**elliptic**motion, which are called perturbations.The simplest method of presenting it starts with the second view of the

**elliptic**motion already set forth.These ever varying elements represent an ever varying

**elliptic**orbit, - not an orbit which the planet actually describes through its whole course, but an ideal one in which it is moving at each instant, and which continually adjusts itself to the actual motion of the planet at the instant.We first conceive of the planets as moving in invariable

**elliptic**orbits, and thus obtain approximate expressions for their positions at any moment.A certain mean

**elliptic**orbit, as near as possible to the actual varying orbit of the planet, is taken.In this orbit a certain fictitious planet is supposed to move according to the law of

**elliptic**motion.This gave the

**elliptic**inequality known as the " equation of the centre," and no other was at that time obvious.The Rudolphine Tables (Ulm, 1627), computed by him from

**elliptic**elements, retained authority for a century, and have in principle never been superseded.Schiaparelli's announcement that the orbit of the bright comet of 1862 agreed strictly with the

**elliptic**ring formed by the circulating Perseid meteors; and three other cases of close coincidence were soon afterwards brought to light.In conformity with the form of the path, the light is said to be elliptically polarized, rightor left-handedly as the case may be, and the axes of the

**elliptic**path are determined by the planes of maximum and minimum polarization of the light.A more pronounced case of

**elliptic**polarization by reflection is afforded by metals.The optical constants (refractive index and co-efficient of extinction) of the metal may then be obtained from observations of the principal incidence and the

**elliptic**polarization then produced.One method consists in finding directly the

**elliptic**constants of the vibration by means of a quarterwave plate and an analyser; but the more usual plan is to measure the relative retardation of two rectangular components of the stream by a Babinet's compensator.Archimedes contributed to the knowledge of these curves by determining the area of the parabola, giving both a geometrical and a mechanical solution, and also by evaluating the ratio of

**elliptic**to circular spaces.Of supreme importance is the fertile conception of the planets revolving about the sun in

**elliptic**orbits.Geographical latitude, which is used in mapping, is based on the supposition that the earth is an

**elliptic**spheroid of known compression, and is the angle which the normal to this spheroid makes with the equator.Some more recent subjects (for example AES and

**elliptic**curve cryptography) are not well covered here.**Elliptic**curves offer major advances on older systems such as increased speed, less memory and smaller key sizes.The leaves are alternate, oval to ovate or

**elliptic**, mostly toothed, rarely entire and occasionally lobed.It is evergreen, and easily recognised by its leathery, bright green, three-nerved leaves,

**elliptic**in shape.The three subjects to which Smith's writings relate are theory of numbers,

**elliptic**functions and modern geometry; but in all that he wrote an "arithmetical" made of thought is apparent, his methods and processes being arithmetical as distinguished from algebraic. He had the most intense admiration of Gauss.The area of the complete curve is 2a 2, and the length of any arc may be expressed in the form f(1 - x 4) - i dx, an

**elliptic**integral sometimes termed the lemniscatic integral.In all the cases which have yet arisen in astronomy the extraneous forces are so small compared with the gravitation of the central body that the orbit is approximately an ellipse, and the preliminary computations, as well as all determinations in which a high degree of precision is not necessary, are made on the hypothesis of

**elliptic**orbits.Let the curve represent an

**elliptic**orbit, AB being the major axis, DE the minor axis, and F the focus in which the centre of attraction is situated, which centre we shall call the sun.The general equation of degree 5 cannot be solved algebraically, but the roots can be expressed by means of

**elliptic**modular functions.**Elliptic**orbits, and a parabolic orbit considered as the special case when the eccentricity of the ellipse is 1, are almost the only ones the astronomer has to consider, and our attention will therefore be confined to them in the present article.Walls, inclined to each other at obtuse angles, enclosed a plot of ground having in the middle a low tumulus of

**elliptic**form, about 35 metres from east to west by 20 from north to south.