# Elliptic Sentence Examples

- In the notation of
**elliptic**integrals. - With this end in view he expounded to the Berlin academy in 1849 a mode of determining an
**elliptic**orbit from three observations, and communicated to that body in 1851 a new method of calculating planetary perturbations by means of rectangular coordinates (republished in W. - 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. - 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. - Long,
**elliptic-ovate**, doubly toothed, pointed, numerously ribbed, hairy below and opaque, and not glossy as in the beech, have short stalks and when young are plaited. - The
**elliptic**lemniscate has for its equation (x 2 +31 2) 2 =a 2 x 2 +b 2 y 2 or r 2 = a 2 cos 2 9 +b 2 sin 20 (a> b). **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.- 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. - Another form, associated with the theory of
**elliptic**functions, has been considered by Dingeldey (Math. - The general equation of degree 5 cannot be solved algebraically, but the roots can be expressed by means of
**elliptic**modular functions. - - Employ the
**elliptic**coordinates n,, and -=n+Vi, such that z=cch?, cchncos,y=cshnsin-; (1) then the curves for which n and are constant are confocal ellipses and hyperbolas, and -d(n,) =c 2 (ch 2 n - cost) = 2c 2 (ch2n-cos2) = r i r 2 = OD 2, (2) if OD is the semi-diameter conjugate to OP, and ri, r 2 the focal distances, rl,r2 = c (ch n cos 0; r 2 = x2 +y2 = c 2 (ch 2 n - sin20 = 1c 2 (ch 2 7 7 +cos 2?). - 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). - Ch (2n +1)I 7 ry /a yl-R3ct (2n +I)3.ch(2n+I)17b /a ' 16 cos(2n+I) 2 7 z /a w1=4,i+ 4, ii = iR ?3a2 an
**elliptic-function**Fourier series; with a similar expression for 1,'2 with x and y, a and b interchanged; and thence 4, = '1 +h. - 2, so that is an
**elliptic**function of t, except when c =a, or 3a. - Put S2 1 =12 cos 4, 12 2 = -12 sin 4, d4 d52 1 dS22 Y a2+c2 122 7Ti = 71 22 CL2- c2(121+5221)J, a2 +c2 do a2+c2 + 4c2 z dt a'-c2 (a2+,c2)2 M+2c2(a2-c2 N-{-a2+c2 2 Ý_a 2 +c 2 (' 4c2 .?"d za 2 -c 2 c2)2 2'J Z M+ -c2) which, as Z is a quadratic function of i 2, are non-
**elliptic**so also for; G, where =co cos, G, 7 7 = - sin 4. - = a constant, so that we may put MdX (17) (a2+X)P' P2= 4(a2 + X)(b2 +X)(c 2 +X), (18) where M denotes a constant; so that 4) is an
**elliptic**integral of th second kind. - 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? ? - A torsion of the ellipsoidal surface will give rise to a velocity function of the form 4)--- where SZ can be expressed by the
**elliptic**integrals in a similar manner, since dX/P3. - In the motion which can be solved by the
**elliptic**function, the most general expression of the kinetic energy was shown by A. - Clebsch to take the form T= 2p(x12 +x22)+2p'x32 + q (xiyi +x2y2) +q'x3y3 +2r(y12+y22)+2r'y32 so that a fourth integral is given by dy 3 /dt = o, y = constant; dx3 (4 y) (q + y) _ (y y) dt - xl 'x2 xl Y Y x l 2 - 1, y2 () = (x12 +x22) (y12 + y22) = (X 1 2 + X 2) +y22)-(FG-x3y3)2 = (x 1 y32-G2)-(Gx3-Fy3) 2, in which 2 = F 2 -x3 2, x l y l +x2y2 = FG-x3y3, Y(y1 2 +y2 2) = T -p(x12 +x22) -p'x32 -2q(xiyi 'x2y2)- 2 q ' x = (p -p') x 2 + 2 (- q ') x 3 y 3+ m 1, (6) m1 = T 2 i y 3 2 (7) so that dt3) 2 =X3, (8) where X3 is a quartic function of x3, and thus t is given by an
**elliptic**(8) (6) (I) integral of the first kind; and by inversion x 3 is in**elliptic**function of the time t. - Now (x1 - x21) (y 1 +y21) = xl l +x2y2 + - (' r 1 2 - x2y1) = FG-x3y3+iV X3, yi+3 7 21_FG-x3y3+2V X3 xl+x21 X12 +X22 (x 1 +x 2 i) = - i{(q' - q)x3+r'y3]+irx3(y1+y21), = FG - x3y3 +ZJ X3 dt2log(x1+x22) - - (q g) x 3- r y3+rx3 F2x32 (12) d dl2 log V x1 ± x2 2 (q'-q)x3-(r'-r) y3FrFF2-x 2 3 ' (13) requiring the
**elliptic**integral of the third kind; thence the expression of x1-f -x21 and yl-}-y21. - Introducing Euler's angles 0, c15, x1= F sin 0 sin 0, x 2 =F sin 0 cos 0, xl+x 2 i =iF sin 0e_, x 3 = F cos 0; sin o t=P sin 4+Q cos 0, dT F sin 2 0d l - dy l + dy 2x = (qx1+ryi)xl +(qx2+ry2)x2 = q (x1 2 +x2 2) +r (xiyi +x2y2) = qF 2 sin 2 0-Fr (FG - x 3 y 3), (16) _Ft (FG _x 323 Frdx3 (17) F x3 X3
**elliptic**integrals of the third kind. - A number of cases are worked out in the American Journal of athematics (1907), in which the motion is made algebraical by the se of the pseudo-
**elliptic**integral. - Sin o= F dl, (20) C3 do F2 h _ F2 cos 2 o F 2 sin z o F dt y - V C G c +2 c1 coso+H]; (21) 1 z so that cos 0 and y is an
**elliptic**function of the time. - 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. - In
**elliptic**integrals, the amplitude is the limit of integration when the integral is expressed in the form f 4) 1% I - N 2 sin e 4) d4. - 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. **ELLIPTICITY**, in astronomy, deviation from a circular or spherical form; applied to the**elliptic**orbits of heavenly bodies, or the spheroidal form of such bodies.- 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. - On 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. - Legendre 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. - Some 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. - Luminous arcs (T), tangential to the upper and lower parts of each halo, also occur, and in the case of the inner halo, the arcs may be prolonged to form a quasi-
**elliptic**halo.1 The physical explanation of halos originated with Rene Descartes, who ascribed their formation to the presence of icecrystals in the atmosphere. - 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. - In a number of cases measures of the relative positions of the two stars, continued for many years, have shown that they are revolving about a common centre; when this is so there can be no doubt that they form a binary system, and that the two components move in
**elliptic**orbits about the common centre of mass, controlled by their mutual gravitation. - How to place a plane quadrilateral of given form so that its geometric shadow may be a square; how to place an
**elliptic**disk, with a small hole in it, so that the shadow may be circular with a bright spot at its centre, &c. - The formation of pictures of the sun in this way is well seen on a calm sunny day under trees, where the sunlight penetrating through small chinks forms
**elliptic**spots on the ground. - The path is therefore an ellipse of which a, b are conjugate semi-diameters, and is described in the period 24 Ju; moreover, the velocity at any point P is equal to ~ OD, where OD is the semi-diameter conjugate to OP. ~,This type of motion;,s called
**elliptic**harmonic. If the co-ordinate axes are the principal axes of the ellipse, the angle ft in (I o) is identical with the excentric angle. - In an
**elliptic**orbit the area irab is swept over in the time irab 22-a r-~---j~ (10) - But since an ellipse can always be constructed with a given centre so as to touch a given line at a given point, and to have a given value of ab(=h/-~ u) we infer that the orbit will be
**elliptic**whatever the initial circumstances. - In
**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. - In a slightly disturbed rapid precession the superposed vibration is
**elliptic-harmonic**, with a period equal to that of the precession itself. - To complete the solution of (2) a third integral is required; this involves in general the use of
**elliptic**functions. - We have, in the notation of
**elliptic**functions, 4= am u. - The most important are: (I) To express the time of describing an
**elliptic**arc under the Newtonian law of gravitation in terms of the focal distances of the initial and final points, and the length of the chord joining them. - - Confocal
**Elliptic**Cylinders. - 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.