The velocity function of the liquid inside the ellipsoid λ = 0 due to the same angular velocity will be
and on the surface outside
|φ0 = xyχ0 = xy||N||B0 − A0||,|
|abc||a2 − b2|
so that the ratio of the exterior and interior value of φ at the surface is
|φ0||=||B0 − A0||,|
|φ1||(a2 − b2) / (a2 + b2) − (B0 − A0)|
and this is the ratio of the effective angular inertia of the liquid, outside and inside the ellipsoid λ = 0.
The extension to the case where the liquid is bounded externally by a fixed ellipsoid λ = λ1 is made in a similar manner, by putting
and the ratio of the effective angular inertia in (9) is changed to
Make c = ∞ for confocal elliptic cylinders; and then
|Aλ = ∫∞λ||ab||=||ab||( 1 − √||b2 + λ||),|
|(a2 + λ) √ (4a2 + λb2 + λ)||a2 − b2||a2 + λ|
|Bλ =||ab||( √||a2 + λ||− 1 ), Cλ = 0;|
|a2 − b2||b2 + λ|
and then as above in § 31, with
the ratio in (11) agrees with § 31 (6).
As before in § 31, the rotation may be resolved into a shear-pair, in planes perpendicular to Ox and Oy.
A torsion of the ellipsoidal surface will give rise to a velocity function of the form φ = xyzΩ, where Ω can be expressed by the elliptic integrals Aλ, Bλ, Cλ, in a similar manner, since
48. The determination of the φ’s and χ’s is a kinematical problem, solved as yet only for a few cases, such as those discussed above.
But supposing them determined for the motion of a body through a liquid, the kinetic energy T of the system, liquid and body, is expressible as a quadratic function of the components U, V, W, P, Q, R. The partial differential coefficient of T with respect to a component of velocity, linear or angular, will be the component of momentum, linear or angular, which corresponds.
Conversely, if the kinetic energy T is expressed as a quadratic function of x1, x2, x3, y1, y2, y3, the components of momentum, the partial differential coefficient with respect to a momentum component will give the component of velocity to correspond.
These theorems, which hold for the motion of a single rigid body, are true generally for a flexible system, such as considered here for a liquid, with one or more rigid bodies swimming in it; and they express the statement that the work done by an impulse is the product of the impulse and the arithmetic mean of the initial and final velocity; so that the kinetic energy is the work done by the impulse in starting the motion from rest.
Thus if T is expressed as a quadratic function of U, V, W, P, Q, R, the components of momentum corresponding are
|x1 =||dT||, x2 =||dT||, x3 =||dT||,|
|y1 =||dT||, y2 =||dT||, y3 =||dT||;|
but when it is expressed as a quadratic function of x1, x2, x3, y1, y2, y3,
|U =||dT||, V =||dT||, W =||dT||,|
|P =||dT||, Q =||dT||, R =||dT||.|
The second system of expression was chosen by Clebsch and adopted by Halphen in his Fonctions elliptiques; and thence the dynamical equations follow
|X =||dx1||− x2||dT||+ x3||dT||, Y = ..., Z = ...,|
|L =||dy1||− y2||dT||+ y3||dT||− x2||dT||+ x3||dT||, M = ..., N = ...,|
where X, Y, Z, L, M, N denote components of external applied force on the body.
These equations are proved by taking a line fixed in space, whose direction cosines are l, m, n, then
|dl||= mR − nQ,||dm||= nP − lR,||dn||= lQ − mP.|
If P denotes the resultant linear impulse or momentum in this direction
|dP||=||dl||x1 +||dm||x2 +||dn||x3|
|+ l||dx1||+ m||dx2||+ n||dx3||,|
|= l (||dx1||− x2R + x3Q )|
|+ m (||dx2||− x3P + x1R )|
|+ n (||dx3||− x1Q + x2P )|
for all values of l, m, n.
Next, taking a fixed origin Ω and axes parallel to Ox, Oy, Oz through O, and denoting by x, y, z the coordinates of O, and by G the component angular momentum about Ω in the direction (l, m, n)
G = l (y1 − x2z + x3y) + m (y2 − x3x + x1z) + n (y3 − x1y + x2x).
Differentiating with respect to t, and afterwards moving the fixed origin up to the moving origin O, so that
|x = y = z = 0, but||dx||= U,||dy||= V,||dz||= W,|
|dG||= l (||dy1||− y2R + y3Q − x2W + x3V )|
|+ m (||dy2||− y3P + y1R − x3U + x1W )|
|+ n (||dy3||− y1Q + y2P − x1V + x2U )|
for all values of l, m, n.
When no external force acts, the case which we shall consider, there are three integrals of the equations of motion
(i.) T = constant, (ii.) x12 + x22+ x32 = F2, a constant, (iii.) x1y1 + x2y2 + x3y3 = n = GF, a constant;
and the dynamical equations in (3) express the fact that x1, x2, x3 are the components of a constant vector having a fixed direction; while (4) shows that the vector resultant of y1, y2, y3 moves as if subject to a couple of components
and the resultant couple is therefore perpendicular to F, the resultant of x1, x2, x3, so that the component along OF is constant, as expressed by (iii).
If a fourth integral is obtainable, the solution is reducible to a quadrature, but this is not possible except in a limited series of cases, investigated by H. Weber, F. Kötter, R. Liouville, Caspary, Jukovsky, Liapounoff, Kolosoff and others, chiefly Russian mathematicians; and the general solution requires the double-theta hyperelliptic function.
49. 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 = 1p (x12 + x22) + 1p′x32+ q (x1y1 + x2y2) + q′x3y3+ 1r (y12 + y22) + 1r′y32
so that a fourth integral is given by
|dx3||= x1 (qx2 + ry2) − x2 (qx1 + ry1) = r (x1y2 − x2y1),|
|1||(||dx3||)||2||= (x12 + x22) (y12 + y22) − (x1y1 + x2y2)2|
= (x12 + x22) (y12 + y22) − (FG − x3y3)2 = (x12 + x22) (y12 + y22 + y32 − G2) − (Gx3 − Fy3)2,
r (y12 + y22) = 2T − p(x12 + x22) − p′x32− 2q (x1y1 + x2y2) − 2q′x3y3 − r′y32= (p − p′) x32 + 2 (q − q′) x3y3 + m1,
where X3 is a quartic function of x3, and thus t is given by an elliptic