## Elliptic Sentence Examples

- In the notation of
**elliptic**integrals. **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.- 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 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. - 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). - 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. - 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. - 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.- 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 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. - 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. - 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.