and then as above in § 31, with
a = c ch α, b = c sh α, a1 = √ (a2 + λ) = c ch α1, b1 = c sh α1
(13)
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
Ω = L ∫∞λ dλ / P3.
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.