This website uses cookies to ensure you get the best experience. Learn more

ellipsoid

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.

11The 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.

00When 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.

00An 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.

00du 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.

00When 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).

00Laplace 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.

00As 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.

002 = o, (8) so that a liquid particle remains always on a similar ellipsoid.

00(17) ellipsoid of liquid of three unequal axes, rotating bodily about the least axis;.

00A system of confocal ellipsoids 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 ellipsoid; and we seek to determine the motion set up, and the form of >G which will satisfy the equation of continuity.

00Over 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?

00= 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.

00l ' 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.

00+4) =0, (19) and this is the infinite boundary ellipsoid if we make the upper limi =co.

00The 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.

00The 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.

00When 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'.

00The extension of Green's solution to a rotation of the ellipsoid was made by A.

00= 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.

00The 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? ?

00The 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-?

00Snakes 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.

00A more difficult case is presented by the ellipsoid.'

00We have first to determine the mode in which electricity distributes itself on a conducting ellipsoid in free space.

00It is a ellipsoid.

00Let 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.

00Thus 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.

00This 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.

00Accordingly 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.

00all 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.

00Hence 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).

00The 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.

00Thus 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.

00Accordingly 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.

00The capacity C of the ellipsoid of revolution is therefore given by the expression I I dx (7) C 2a ?

00In each case we have C = a when e= 0, and the ellipsoid thus becomes a sphere.

00In 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.

00The 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.

00If 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.

00Hence the geoid or figure of the sea-surface is not part of an ellipsoid of rotation but is irregular.

00During 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.

0035 The Ellipse and the Ellipsoid.

00- For elementary mensuration the ellipse is to be regarded as obtained by projection of the circle, and the ellipsoid by projection of the sphere.

00Hence 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.

00This formula applies to such figures as the cone, the sphere, the ellipsoid and the prismoid.

00Monge'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.

00showed 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.

00When 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.

00The 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.

00If 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)

00The ellipsoid (26) was first employed by J.

00Binet (1811), and may be called Binets Ellipsoid for the point 0.

00It 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)

00If 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)

00for different values of 0 this represents a system of quadrics confocal with the ellipsoid ~f+~1+~I, (~4)

00which we shall meet with presently as the ellipsoid of gyration at G.

00Since they are essentially positive the quadric is an ellipsoid; it is called the momental ellipsoid at 0.

00A limitation is thus imposed on the possible forms of the momental ellipsoid; e.g.

00in 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.

00If 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)

00This is called the ellipsoid of gyration at 0; it was introduced into the theory by J.

00If 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.

00The relation between these axes may be expressed by means of the momental ellipsoid at 0.

00If p be the radius-vector 0J of the momental ellipsoid Ax+By+Czf=Me4 (I)

00We 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)

00The 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.

00It has been shown by Dc Sparre that, owing to the limitation imposed on the possible forms of the momental ellipsoid by the relation B+C>A, the curve has no points of inflexion.

00Vegetative cells cylindric (rodlets), ellipsoid or ovoid, and straight.

00Ellipsoid >>

00ellipsoid of revolution, the following graph shows how the strain energy is related to the shape.

00ellipsoid defined by the error radii centered on P1.

00ellipsoid designed to fit the shape of the entire earth.

00UTM public UTM (ellipsoid ellipsoid, float lat, float lng) Initialises the UTM converter with the given ellipsoid.

00The principal axes of the thermal ellipsoid can be obtained from the U values via a principal axes transformation.

00These atoms lie at each end of the principal axis of the original atoms anisotropic adp ellipsoid.

00Then, the superquadric ellipsoid is a useful object.

00A mathematical ellipsoid designed to fit the shape of the entire earth.

00ellipsoid data to be placed.

00ellipsoid surface by a number of meters.

00ellipsoid shape that can be used to model a cloud pretty good.

00ellipsoid plots containing custom atom colors became truncated during on-screen rotation.

00ellipsoid parameters can now be safely duplicated.

00ellipsoid settings can be altered (see Section 6.4 ).

00Height is the elevation of the mean surface observed at nadir above the reference ellipsoid.

00Add the error ellipsoids of each shot to get an error ellipsoid for the entire loop.

00Principal strains, strain ellipsoid and strain ellipse; special types of strain ellipsoid and Flinn plots.

00jigger factor, N, must be added to the basic ellipsoid radius.

00UTM public UTM (Ellipsoid ellipsoid, float lat, float LNG) Initialises the UTM converter with the given ellipsoid.

00plane of the equator of the ellipsoid.

00In 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.

00The 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.

00When 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.

00An 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.

00The 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).

00du 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.

00When 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).

00Laplace 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.

00As 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.

002 = o, (8) so that a liquid particle remains always on a similar ellipsoid.

00(17) ellipsoid of liquid of three unequal axes, rotating bodily about the least axis;.

00The 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.

00A system of confocal ellipsoids 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 ellipsoid; and we seek to determine the motion set up, and the form of >G which will satisfy the equation of continuity.

00Over 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?

00= 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.

00l ' 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.

00+4) =0, (19) and this is the infinite boundary ellipsoid if we make the upper limi =co.

00The 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.

00The 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.

00When 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'.

00The extension of Green's solution to a rotation of the ellipsoid was made by A.

00= 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.

00The 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? ?

00The 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-?

00Snakes 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.

00A more difficult case is presented by the ellipsoid.'

00We have first to determine the mode in which electricity distributes itself on a conducting ellipsoid in free space.

00It is a ellipsoid.

00over an ellipsoid, so that its density is everywhere proportional to the thickness of a shell formed by describing round ' The solution of the problem of determining the distribution en an ellipsoid of a fluid the particles of which repel each other with a force inversely as the nth power of the distance was first given by George Green (see Ferrer's edition of Green's Collected Papers, p. 119, 1871).

00Let 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.

00Thus 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.

00This 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.

00Accordingly 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.

00all 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.

00Hence 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).

00The 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.

00Thus 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.

00Accordingly 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.

00The capacity C of the ellipsoid of revolution is therefore given by the expression I I dx (7) C 2a ?

00(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).

00In each case we have C = a when e= 0, and the ellipsoid thus becomes a sphere.

00In 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.

00The 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.

00If 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.

00Hence the geoid or figure of the sea-surface is not part of an ellipsoid of rotation but is irregular.

00During 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.

0035 The Ellipse and the Ellipsoid.

00- For elementary mensuration the ellipse is to be regarded as obtained by projection of the circle, and the ellipsoid by projection of the sphere.

00Hence 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.

00This formula applies to such figures as the cone, the sphere, the ellipsoid and the prismoid.

00Monge'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.

00showed 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.

00When 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.

00The 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.

00If 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)

00The ellipsoid (26) was first employed by J.

00Binet (1811), and may be called Binets Ellipsoid for the point 0.

00It 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)

00If 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)

00for different values of 0 this represents a system of quadrics confocal with the ellipsoid ~f+~1+~I, (~4)

00which we shall meet with presently as the ellipsoid of gyration at G.

00Since they are essentially positive the quadric is an ellipsoid; it is called the momental ellipsoid at 0.

00A limitation is thus imposed on the possible forms of the momental ellipsoid; e.g.

00in 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.

00If 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)

00This is called the ellipsoid of gyration at 0; it was introduced into the theory by J.

00If 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.

00If 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.

00The relation between these axes may be expressed by means of the momental ellipsoid at 0.

00If p be the radius-vector 0J of the momental ellipsoid Ax+By+Czf=Me4 (I)

00We 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)

00The 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.

00It has been shown by Dc Sparre that, owing to the limitation imposed on the possible forms of the momental ellipsoid by the relation B+C>A, the curve has no points of inflexion.

00Vegetative cells cylindric (rodlets), ellipsoid or ovoid, and straight.

00