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 dT dT dT x1 = --, x2 = --, x3 = --, (1) dU dV dW dT dT dT y1 = --, y2 = --, y3 = --; dP dQ dR but when it is expressed as a quadratic function of x1, x2, x3, y1, y2, y3, dT dT dT U = ---, V = ---, W = ---, (2) dx1 dx2 dx3 dT dT dT P = ---, Q = ---, R = ---. 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 dx1 dT dT X = --- - x2--- + x3---, Y = ..., Z = ..., (3) dt dy3 dy2 dy1 dT dT dT dT L = --- - y2--- + y3--- - x2--- + x2---, M = ..., N = ..., (4) dt dy3 dy2 dx3 dx2 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 dm dn -- = mR - nQ, -- = nP - lR, -- = lQ - mP. (5) dt dt dt If P denotes the resultant linear impulse or momentum in this direction P = lx1 + mx2 + nx3, (6) dP dl dm dn -- = --x1 + --x2 + --x3 dt dt dt dt dx1 dx2 dx3 + l--- + m--- + n---, dt dt dt / dx1 \ = l ( --- - x2R + x3Q ) \ dt / / dx2 \ + m ( --- - x3P + x1R ) \ dt / / dx3 \ + n ( --- - x1Q + x2P ) \ dt / = lX + mY + nZ, (7) for all values of l, m, n. Next, taking a fixed origin [Omega] 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 [Omega] in the direction (l, m, n) G = l(y1 - x2z + x3y) + m(y2 - x3x + x1z) + n(y3 - x1y + x2x). (8) Differentiating with respect to t, and afterwards moving the fixed origin up to the moving origin O, so that dx dy dz x = y = z = 0, but -- = U, -- = V, -- = W, dt dt dt