Free Solids through a Liquid. 365 



[u, u] == [v, v] = [w,w~\ (let m be their common value), 

 [©•,<*] = [p,p] = [<r,<r] „ n „ „ „ 



[u,vr] = [v ) p-\=:[w > (T] „ h „ „ „ 



\v, w] = [w, u] = \u } v] = 0; [p,a] = [<7,*r] = [cr,f>] =0^ 



and 



(10) 



[u, p] = [w,<r] = [v, <r] = [w, or] = [to, w] = [w, p] = ; J 

 s o that the formula for T is 



T = i{m(u 2 + v Q + w 2 )+n{& 2 + p 2 + a2)+2h(u*T + vp + w<T)} (11) 

 For this case, therefore, the Eulerian equations (1) become 



d(mu + hvr) . . Y « 



— — -i- — m{v<7 — wp) = JL i &c, 



and 



d{nm + hu) 



= L, &c. 



>- • (11) 



[Memorandum : — Lines of reference fixed relatively 

 to the body.] 



But inasmuch as (11) remains unchanged when the lines of 

 reference are altered to any other three lines at right angles to 

 one another through P, it is easily shown directly from (6), (7), 

 and (9) that if, altering the notation, we take u, v, w to denote 

 the components of the velocity of P parallel to three fixed rectan- 

 gular lines, and ot, p, cr the components of the body's angular 

 velocity round these lines, we have 



and 



d(mu -h h'si) 

 It 



d{nw + hu) 



=X, &c, 



+ h(crv — pw) = Jj, &c. 



^ 



(12) 



[Memorandum : — Lines of reference fixed in space] , 

 which are more convenient than the Eulerian equations. 



The integration of these equations, when neither force nor 

 couple acts on the body (X = &c, L = &c), presents no 

 difficulty; but its result is readily seen from § 21 (" Vortex 

 Motion ") to be that, when the impulse is both translatory and 

 rotational, the point P, round which the body is isotropic, 

 moves uniformly in a circle or spiral so as to keep at a constant 

 distance from the " axis of the impulse/' and that the compo- 

 nents of angular velocity round the three fixed rectangular axes 

 are constant. 



