Some importance attaches to the form of the pollen grains; the two principal forms are ellipsoidal with longitudinal bands forming the Convolvulus-type, and a spherical form with a spiny surface known as the Ipomaea-type.
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.
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.
The quiescent ellipsoidal surface, over which the motion is entirely tangential, is the one for which (a2+X)d?
Clebsch, by taking a velocity function 4,=xyx (I) for a rotation R about Oz; and a similar procedure shows that an ellipsoidal surface A may be in rotation about Oz without disturbing the motion if I I dx + _ a2'-A) x 2 a R t i/(b2+A)- i/(a2+A) and that the continuity of the liquid is secured if (a 2 _ I -A) 3/2 (b 2 4 A)3f2(C2 -+- A) 2 ??
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.
(9) c 2 Ci If the shot is moving as if fired from a gun of calibre d inches, in which the rifling makes one turn in a pitch of n calibres or nd inches, so that the angle S of the rifling is given by tan S = ird/nd = 2 d p/u, (10) '°If a denotes the density of the metal, and if the shell has a cavity homothetic with the external ellipsoidal shape, a fraction f of the linear scale; then the volume of a round shot being sird 3, and sird 3 x of a shot x calibres long W =*ird 3 x(I -f 3)v, (20) 2 Wki 2= 61rd 3 xo(I-f 5)Q, (21) Wk22=67rd3x 2 2+0 2(I - f5)Q.
2 Consider then an ellipsoidal shell the axes of whose bounding surfaces are (a, b, c) and (a+da), (b+db), (c+dc), where da/a =db b =dc/c =,u.
The Cassegrain telescope differs from the Gregorian only in the substitution of a convex hyperboloidal mirror for a concave ellipsoidal mirror as the small speculum.
Ellipsoidal, pear-shaped or hour-glassshaped stars would all give rise to the phenomena of a short-period variable, and doubtless examples of these intermediate forms exist.
By its aid, for example, the whole of the properties a elliptical arches, whether square or skew, whether level or sloping in their span, are at once deduced by projection from those of symmetrical circular arches, and the properties of ellipsoidal and ellipticconoidal domes from those of hemispherical and circular-conoidal domes; and the figures of arches fitted to resist the thrust of earth, which is less horizontally than vertically in a certain given ratio, can be deduced by a projection from those of arches fitted to resist the thrust of a liquid, which is of equal intensity, horizontally and vertically.
The bright orangeyellow fruit is round or ellipsoidal, about i in.
The residual force just described is not limited to the case of an ellipsoidal body.