dG / dy1 \ -- = l ( --- - y2R + y3Q - x2W + x3V ) dt \ dt / / dy2 \ + m ( --- - y3P + y1R - x3U + x1W ) \ dt / / dy3 \ + n ( --- - y1Q + y2P - x1V + x2U ) \ dt / = lL + mM + nN, (9) 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 x2W - x3V, x3U - x1W, x1V - x2U, (10) 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 = ½p(x1² + x2²) + ½p´x3² + q(x1y1 + x2y2) + q´x3y3 + ½r(y1² + y2²) + ½r´y3² (1) so that a fourth integral is given by dy3/dt = 0, y3 = constant; (2) dx3 --- = x1(qx2 + ry2) - x2(qx1 + ry1) = r(x1y2 - x2y1), (3) dt 1 / dx3 \² -- ( --- ) = (x1² + x2²)(y1² + y2²) - (x1y1 + x2y2)² r² \ dt / = (x1² + x2²)(y1² + y2²) - (FG - x3y3)² = (x1² + x2²)(y1² + y2² + y3² - G²) - (Gx3 - Fy3)², (4) in which x1² + x2² = F² - x3², x1y1 + x2y2 = FG - x3y3, (5) r(y1² + y2²) = 2T - p(x1² + x2²) - p´x3² - 2q(x1y1 + x2y2) - 2q´x3y3 - r´y3² = (p - p´)x3² + 2(q - q´)x3y3 + m1, (6) m1 - 2T - pF² - 2qFG - r1y3² (7) so that 1 / dx3 \² -- ( --- ) = X3 (8) r² \ dt / where X3 is a quartic function of x3, and thus t is given by an elliptic integral of the first kind; and by inversion x3 is in elliptic function of the time t. Now (x1 - x2i)(y1 + y2i) = x1y1 + x2y2 + i(x1y2 - x2y1) = FG - xy3y3 + i[V-]X3, (9) y1 + y2i FG - x3y3 + i[root]X3 -------- = --------------------- , (10) x1 + x2i x1² + x2² d -- (x1 + x2i) = -i[(q´ - q)x3 + r´y3] + irx3(y1 + y2i), (11) dt