# 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 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. - Du Bois (Magnetic Circuit, p. 33), the demagnetizing factor, and the ratio of the length of the
**ellipsoid**2c to its equatorial diameter 2a (=c/a), the dimensional ratio, denoted by the symbol nt. - 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. - (17)
**ellipsoid**of liquid of three unequal axes, rotating bodily about the least axis;. - A system of confocal
is taken y2 (3) a 2 +X b 2 +X c2 + A= I, and a velocity function of the form = x1 P, (4) where 4' is a function of X only, so that 4) is constant over an**ellipsoids****ellipsoid**; and we seek to determine the motion set up, and the form of >G which will satisfy the equation of continuity. - 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? - = dx ?+xd%y ds ds ds ds +2 l dd, so that the velocity of the liquid may be resolved into a component -41 parallel to Ox, and -2(a 2 +X)ld4/dX along the normal of the
**ellipsoid**; and the liquid flows over an**ellipsoid**along a line of slope with respect to Ox, treated as the vertical. - L ' so that over the surface of an
**ellipsoid**where X and ¢ are constant, the normal velocity is the same as that of the**ellipsoid**itself, moving as a solid with velocity parallel to Ox U = -q, - 2 (a2+X) dtP, and so the boundary condition is satisfied; moreover, any**ellipsoidal**surface X may be supposed moving as if rigid with the velocity in (I I), without disturbing the liquid motion for the moment. - +4) =0, (19) and this is the infinite boundary
**ellipsoid**if we make the upper limi =co. - The velocity of the
**ellipsoid**defined by X =o is then U= - 2 __ M ((ro b J o (a2 =ab (i -A0), (20) with the notation A or A a a= a (a2bc+ = - 2abc d -- so that in (4) xA x 'UxA x A' 4)' 1 -Ao' (22) in (I) for an**ellipsoid**. - The impulse required to set up the motion in liquid of density p i the resultant of an impulsive pressure p4) over the surface S of th
**ellipsoid**, and is therefore ffp4ldS = p4GoffxldS =p 40 (volume of the**ellipsoid**) =4)oW', (23) where W' denotes the weight of liquid displaced. - 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 of Green's solution to a rotation of the
**ellipsoid**was made by A. - = constant, _ ff 00 NdA N BA-AA X - JA (a' +X) (b 2 +A)P - abc' a2 -b2 ' and at the surface A = o, I I N Bo-A 0 N I R - (a2+b2) abc a 2 -b 2 abc a2b2 I /b 2 N = R I /b2 - I /a2 abc I 1 I Bo - AO' a 2 b 2 - a2 b2 a 2 b2 = R (a 2 - b 2) /(a 22 + /b2) 2 - r (B o - Ao) U Bo+Co - B I - CI' Since - Ux is the velocity function for the liquid W' filling the
**ellipsoid**A = o, and moving bodily with it, the effective inertia of the liquid in the interspace is Ao+B1+C1 Bo+Co - B1 - C, If the**ellipsoid**is of revolution, with b=c, - 2 XBo - - C BI' and the Stokes' current function 4, can be written down (I) is (5) (7) (6) The velocity function of the liquid inside the**ellipsoid**A=o due to the same angular velocity will be = Rxy (a2 - b2)/(a2 + b2), (7) and on the surface outside _ N Bo -Ao c1)0xy abc 2 62' so that the ratio of the exterior and interior value of at the surface is ?o= Bo-Ao (9) 4)1 (a 2 -6 2)/(a2 + b) - (Bo - Ao)' and this is the ratio of the effective angular inertia of the liquid, outside and inside the**ellipsoid**X = o. - 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? ? - The
**ellipsoid**is the only shape for which a and (3 have so far been determined analytically, as shown already in § 44, so we must restrict our calculation to an egg-shaped bullet, bounded by a prolate**ellipsoid**of revolution, in which, with b =c, Ao= fo (a2 + X)V [4(a2+X)(b +X)2]-J0 2(a2 +X)3/2(b2+X), (13) Ao+2Bo = I, (t4) _ B 0 t - A 0 I a?I-A0' Q I - Bo I-{- A o I-? - 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**.' - We have first to determine the mode in which electricity distributes itself on a conducting
**ellipsoid**in free space. - 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. - Accordingly 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. - All from thhiseweecan of determine the capacity of the
**ellipsoid**as follows: Let p be the length of the perpendicular from the centre of the**ellipsoid**, whose equation is x 2 /a 2 -1-y2/b2 -1-,2c2 = i to the tangent plane at x, y, z. - 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. - The capacity C of the
**ellipsoid**of revolution is therefore given by the expression I I dx (7) C 2a ? - (x2+3,2) If the
**ellipsoid**is one of revolution round the major axis a (prolate) and of eccentricity e, then the above formula reduces to I I l og e (I +e) C - tae Whereas if it is an**ellipsoid**of revolution round the minor axis b (oblate), we have I sin - tae C2 - ae (9). - 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. - 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. - Hence the geoid or figure of the sea-surface is not part of an
**ellipsoid**of rotation but is irregular. - - For elementary mensuration the ellipse is to be regarded as obtained by projection of the circle, and the
**ellipsoid**by projection of the sphere. - 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. - Showed that, if the large mirror were a segment of a paraboloid of revolution whose focus is F, and the small mirror an
**ellipsoid**of revolution whose foci are F and P respectively, the resulting image will be plane and undistorted. - 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. - The method of homogeneous strain can be applied to deduce the corresponding results for an
**ellipsoid**of semi-axes a, b, c. If the co-ordinate axes coincide with the principal axes, we find l0=1/2Ma2, I9=~Mb2, I~ = ~ Me2, whence Ii.~ =3/4M (b1 +ci), &c. - If the co-ordinate axes coincide with the principal axes of this quadric, we shall have ~(myz) =0, ~(mzx) =0, Z(mxy) = 0~ (24) and if we write ~(mx) = Ma, ~(my1) = Mb, ~(mz) =Mc2, (25) where M=~(m), the quadratic moment becomes M(aiX2+bI,s2+ cv), or Mp, where p is the distance of the origin from that tangent plane of the
**ellipsoid**~-,+~1+~,=I, (26) - The
**ellipsoid**(26) was first employed by J. - Binet (1811), and may be called Binets
**Ellipsoid**for the point 0. - It may further be shown that if Binets
**ellipsoid**be referred to any system of conjugate diameters as co-ordinate axes, its equation will be ~2+~2+~-2I, (27) - If a, b, c be the semi-axes of the Binets
**ellipsoid**of G, the quadratic moment with respect to the plane Xx + ~iy + vz =0 will be M(aX + bu + c2vi), and that with respect to a parallel plane ?.x+uy+vz=P (29) - For different values of 0 this represents a system of quadrics confocal with the
**ellipsoid**~f+~1+~I, (~4) - Which we shall meet with presently as the
**ellipsoid**of gyration at G. - 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. - In the case of symmetry about an axis it appears that the ratio of the polar to the equatorial diameter of the
**ellipsoid**cannot be less than I/~2. - If we write A=Ma, B=M/32, C=M~y, the formula (37), when referred to the principal axes at 0, becomes if p denotes the perpendicular drawn from 0 in the direction (X, u, e) to a tangent plane of the
**ellipsoid**~+~+~=I (43) - This is called the
**ellipsoid**of gyration at 0; it was introduced into the theory by J. - 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) - We have seen (~ 18) that this vector coincides in direction with the perpendicular OH to the tangent plane of the momental
**ellipsoid**at J; also that ~ (2)