# Ellipsoid Sentence Examples

In this he showed that a homogeneous fluid mass revolving uniformly round an axis under the action of gravity ought to assume the form of an

**ellipsoid**of revolution.The cases of greatest practical importance are those of a sphere (which is an

**ellipsoid**with three equal axes) and an ovoid or prolate**ellipsoid**of revolution.When the

**ellipsoid**is so much elongated that I is negligible in relation to m'-, the expression approximates to the simpler form N=412 (log 201-I).Laplace treated the subject from the point of view of the gradual aggregation and cooling of a mass of matter, and demonstrated that the form which such a mass would ultimately assume must be an

**ellipsoid**of revolution whose equator was determined by the primitive plane of maximum areas.As an application of moving axes, consider the motion of liquid filling the ellipsoidal case 2 y 2 z2 Ti + b1 +- 2 = I; (1) and first suppose the liquid be frozen, and the

**ellipsoid**l3 (4) (I) (6) (9) (I o) (II) (12) (14) = 2 U ¢ 2, (15) rotating about the centre with components of angular velocity, 7 7, f'; then u= - y i +z'i, v = w = -x7 7 +y (2) Now suppose the liquid to be melted, and additional components of angular velocity S21, 522, S23 communicated to the ellipsoidal case; the additional velocity communicated to the liquid will be due to a velocity-function 2224_ - S2 b c 6 a 5 x b2xy, as may be verified by considering one term at a time.Over the

**ellipsoid**, p denoting the length of the perpendicular from the centre on a tangent plane, px _ pv _ _ pz 1= a2+X' b +A' n c2+A p2x2 + p2y2 p2z2 I (a2 - + X)2 (b 2 +x)2 + (0+X)2, p 2 = (a2+A)12+(b2+X)m2+(c2+X)n2, = a 2 1 2 +b 2 m 2 +c 2 n 2 +X, 2p d = ds; (8) Thence d?When the liquid is bounded externally by the fixed

**ellipsoid**A = A I, a slight extension will give the velocity function 4 of the liquid in the interspace as the**ellipsoid**A=o is passing with velocity U through the confocal position; 4 must now take the formx(1'+N), and will satisfy the conditions in the shape CM abcdX ¢ = Ux - Ux a b x 2+X)P Bo+CoB I - C 1 (A 1 abcdX, I a1b1cl - J o (a2+ A)P and any'confocal**ellipsoid**defined by A, internal or external to A=A 1, may be supposed to swim with the liquid for an instant, without distortion or rotation, with velocity along Ox BA+CA-B 1 -C1 W'.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? ?Snakes are oviparous; they deposit from ten to eighty eggs of an

**ellipsoid**shape, covered with a soft leathery shell, in places where they are exposed to and hatched by moist heat.A more difficult case is presented by the

**ellipsoid**.'AdvertisementWe have first to determine the mode in which electricity distributes itself on a conducting

**ellipsoid**in free space.It is a

**ellipsoid**.Let a charge +Q be f t the

**ellipsoid**a similar and slightly larger one, that distribution will be in equilibrium and will produce a constant potential throughout the interior.Thus if Q is the surface density, S the thickness of the shell at any point, and p the assumed volume density of the matter of the shell, we have v =Abp. Then the quantity of electricity on any element of surface dS is A times the mass of the corresponding element of the shell; and if Q is the whole quantity of electricity on the

**ellipsoid**, Q =A times the whole mass of the shell.This mass is equal to 47rabcp,u; therefore Q = A47rabcp s and b =pp, where p is the length of the perpendicular let fall from the centre of the

**ellipsoid**on the tangent plane.AdvertisementAccordingly for a given

**ellipsoid**the surface density of free distribution of electricity on it is everywhere proportional to the the tangent e plane e att that point.Hence the density v is given by 47rabc (x2/a4+y2/b4-I-z2/c4), and the potential at the centre of the

**ellipsoid**, and therefore its potential as a whole is given by the expression, adS Q dS V f r 47rabc r' (x2/a4-I-y2/b4+z2/c4) Accordingly the capacity C of the**ellipsoid**is given by the equation 1 I J dS C 47rabc Y (x 2 +y 2 + z2) V (x2/a4+y2/b4+z2/c4) (5) It has been shown by Professor Chrystal that the above integral may also be presented in the form,' foo C 2 J o J { (a2 + X) (b +X) (c 2 + X) } (6).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.Thus if the

**ellipsoid**is one of revolution, and ds is an element of arc which sweeps out the element of surface dS, we have dS = 27ryds = 27rydx/ (Ts) = 27rydx/ (b y) = 2 p2 dx.Accordingly the distribution of electricity is such that equal parallel slices of the

**ellipsoid**of revolution taken normal to the axis of revolution carry equal charges on their curved surface.AdvertisementThe capacity C of the

**ellipsoid**of revolution is therefore given by the expression I I dx (7) C 2a ?In each case we have C = a when e= 0, and the

**ellipsoid**thus becomes a sphere.In the extreme case when e=1, the prolate

**ellipsoid**becomes a long thin rod, and then the capacity is given by C 1 = a/log e 2a/b (io), which is identical with the formula (2) already obtained.The reader is also referred to an article by Lord Kelvin (Reprint of Papers on Electrostatics and Magnetism, p. 178), entitled " Determination of the Distribution of Electricity on a Circular Segment of a Plane, or Spherical Conducting Surface under any given Influence," where another equivalent expression is given for the capacity of an

**ellipsoid**.If the whole globe were covered with a uniformly deep ocean, and if there were no difference of density between one part and another, the surface would form a perfect

**ellipsoid**of revolution, that is to say, all the meridians would be exactly equal ellipses and all parallels perfect circles.AdvertisementHence the geoid or figure of the sea-surface is not part of an

**ellipsoid**of rotation but is irregular.During Nansen's expedition on the " Fram " in 1894-1895, Scott Hansen made observations with a Sterneck's half-seconds pendulum on the ice where the sea was more than 1600 fathoms deep and found only an insignificant deviation from the number of swings corresponding to a normal

**ellipsoid**.Hence the area of an ellipse whose axes are 2a and 2b is Trab; and the volume of an

**ellipsoid**whose axes are 2a, 2b and 2c is t rabc. The area of a strip of an ellipse between two lines parallel to an axis, or the volume of the portion (frustum) of an**ellipsoid**between two planes parallel to a principal section, may be found in the same way.This formula applies to such figures as the cone, the sphere, the

**ellipsoid**and the prismoid.Monge'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.When the angular momentum is too great for the usual spheroidal form to persist, this gives place to an

**ellipsoid**with three unequal axes; this is succeeded by a pear-shaped form.Binet (1811), and may be called Binets

**Ellipsoid**for the point 0.Since they are essentially positive the quadric is an

**ellipsoid**; it is called the momental**ellipsoid**at 0.A limitation is thus imposed on the possible forms of the momental

**ellipsoid**; e.g.If all the masses lie in a plane (1=0) we have, in the notation of (25), c2 = o, and therefore A = Mb, B = Ma, C = M (a +b), so that the equation of the momental

**ellipsoid**takes the form b2x2+a y2+(a2+b2) z1=s4.The relation between these axes may be expressed by means of the momental

**ellipsoid**at 0.If p be the radius-vector 0J of the momental

**ellipsoid**Ax+By+Czf=Me4 (I)The motion of the body relative to 0 is therefore completely represented if we imagine the momental

**ellipsoid**at 0 to roll without sliding on a plane fixed in space, with an angular velocity proportional at each instant to the radius-vector of the point of contact.The principal axes of the thermal

**ellipsoid**can be obtained from the U values via a principal axes transformation.These atoms lie at each end of the principal axis of the original atoms anisotropic adp

**ellipsoid**.Then, the superquadric

**ellipsoid**is a useful object.A mathematical

**ellipsoid**designed to fit the shape of the entire earth.Height is the elevation of the mean surface observed at nadir above the reference

**ellipsoid**.Add the error ellipsoids of each shot to get an error

**ellipsoid**for the entire loop.Principal strains, strain

**ellipsoid**and strain ellipse; special types of strain**ellipsoid**and Flinn plots.The magnetization at any point inside the

**ellipsoid**will then be I = HN (29) where N=47r (e2t) (-2-eloI- e - t), e being the eccentricity (see Maxwell's Treatise, § 438).The

**ellipsoid**was the shape first worked out, by George Green, in his Research on the Vibration of a Pendulum in a Fluid Medium (2833); the extension to any other surface will form an important step in this subject.If A = B = C, the momental

**ellipsoid**becomes a sphere; all axes through 0 are then principal axes, and the moment of inertia is the same for each.When it is desired to have a uniform magnet with definitely situated poles, it it usual to employ one having the form of an ovoid, or elongated

**ellipsoid**of revolution, instead of a rectangular or cylindrical bar.An important instance in which the calculation can be made is that of an elongated

**ellipsoid**of revolution placed in a uniform field H o, with its axis of revolution parallel to the lines of force.