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 | , |
| dU | dV | dW |
(1)
| y1 = | dT | , y2 = | dT | , y3 = | dT | ; |
| dP | dQ | dR |
but when it is expressed as a quadratic function of x1, x2, x3, y1, y2, y3,
| U = | dT | , V = | dT | , W = | dT | , |
| dx1 | dx2 | dx3 |
(2)
| P = | dT | , Q = | dT | , R = | dT | . |
| dy1 | dy2 | dy3 |
The second system of expression was chosen by Clebsch and adopted by Halphen in his Fonctions elliptiques; and thence the dynamical equations follow