456 Differentiating with respect to the time, and supposing after wards that the centre of the body and the fixed origin are coinci- dent, then, since d *=u , -t, dt at HYDROMECHANICS [HYDRODYNAMICS. Again, put p= a cos g , q - a sin g ; then from (4) and (5) (pq -pq) = (c 4 - c 6 )o-V + (c } - c%}(up + vq)w ; ~ at butse-O, y = 0, 2 = 0, ~dt dT j dv + m d/dT dT dT -Ji -T )- r ^~ + 9T ~ dt dp) dq dr d/dT dT dT dT -TT I -r- I -p~r + T J M-r~ dtdq / r dr dp dw didT dT I - f> ( _T_ dT dt dr ) *dp dq du for all values of Z, m, n. If no external forces act, then three integrals of the equations of motion are (1) T = constant; (r/T 2 /r ?)+( du) </T 2 ?) -constant; dw ) . dT dT T dT dT dT (3) . -- 1 ------- 1 ----- = constant; die dp do dq dw dr expressing the fact- that the energy is constant, and also the force and couple constituents of the resultant impulse. For a body like an ellipsoid, using single suffixes, T = (cjZi 2 + e.. 2 v- + c 3 iv- + c 4 ?r + c,? 2 + c 6 r 2 ) ; to hyperelliptic and double functions (Weber, Annalcn, vol. xiv. ). The equations of motion become MathcmatiscJie du, c w _ (1), 1 dt * (2), dw f. (3); dt dp = . . (4), c dq_ _ . _ c . t = . . (5), dr c s 777 ( C 4 ~ C 5>P1 ~ ( c i ~ c t) u L = . . (6). 2 dg_ a rational function of w ; and therefore g is expressed in terms of the time by elliptic integrals of the third kind. In a state of steady motion, w is constant, and -J- - etc up + vq = S(r ; and therefore ?; also dt ._ JL = ^3- .?L + u__5 _ c. w c, s and we must therefore have r 2 for the roots of this quadratic in s : <r to be real. If we employ the Lagrangian coordinates x, y, z, 6, <, if, and take OZ in the direction of the resultant linear impulse F, then Multiplying the equations by , r, w, p, q, r, and adding, du do dw dp dq dr A Multiply (1) by (11); c ?- 2 ) = T, a constant . . (7). i, (2) by c. 2 ii, and (3) by c.^w, and add ; then da dv dw _r, GI tl dt C * l dt 3 H dt ~~ and c 1 2 w 2 + <y ! i; 2 + c 3 2 M; 2 = F 2 (8), F being a constant, the resultant linear impulse of the motion. Again, multiplying the equations (1 to 6) by c 4 p, c 5 q, C 6 r, c^U, c.ji , c 3 tv, adding and integrating, c l c 4 up + c. 2 c b vq + c A ( 6 icr=G, a constant . . . (9). Equations (4), (5), (6) show that the body is acted upon by com ponent couples about the principal axes (c,-c..)viv, (C. A - CI }WU, behlf , supl10se d to be the equatorial moment of inertia of the "^ -c.,)ni-, the principal moments of inertia being supposed to be I i> O( i v If OT be the direction of motion of 0, then OT lies in the plane Fig. 10. (fig. 10) the eye being supposed at the centre of the sphere, CjM = component momentum in direction OA= - F sin cos0 , CjV=* ,, ,, 013 = F sin sin <p , C A W = ,, ,, 00= F cos ; and therefore equation (10) gives cos as an elliptic function of/. Since jj = sin (f>0 - sin cos <<j/, g = cos <j>0 + sin sin <pty, equation (9) becomes db G - c R r cos or -^= ~ ;,- dt c 4 sin -9 = ~2 <T t 1 - cos 8 2 c 4 1 + cos and therefore ^ will consist of elliptic integrals of the third kind. Equations (4), (5), (6). show that the body is acted upon at every instant by a couple wliose axis is OE, of magnitude Ci> Cjj C 6- If the body be of revolution, c t = c. 2 and c 4 = c a , the motion can be expressed by elliptic functions. For equation (6) shows that r is constant, and equation (3) becomes - (up + vq}* ZOO, and tan = -^i tan 6. c, . .(10), a biquadratic function of ic, and therefore w is an elliptic function of t, the time. Put u = s cos/, v= -s sin/; thin from (1) and (2) We may determine the steady motion from elementary reasoning; for if OG be the axis of the resultant angular momentum (also lying in the plane ZOO) making an angle with 00, and if /* be the con stant value of ij/, then O/t sin (0 - y8) = impressed couple But and therefore or M - / dt a rational function of w ; and therefore / is expressed in terms of v.-Oli 00, tan 7= ?- sin 0. the time by elliptic integrals of the third kind. r = c * (cj - c 3 )tt <2 tan 0. G cos j3 = c (j ? . G sin |3= -c 4 ^ ;
- = - fj. sin ;
tan B = ^- sin ;
and if 7 be the angle made by the axis of instantaneous rotation
