Free Solids through a Liquid. 367 



with (13), used in (6) give the equations of the solid's motion re- 

 ferred to fixed rectangular axes. They have the inconvenience 

 of the coefficients [u, u], [i«, t>], &c. being functions of the an- 

 gular coordinates of the solid. The Eulerian equations (free 

 from this inconvenience) are readily found on precisely the same 

 plan as that adopted above for the old case of no cyclic motion 

 in the fluid. 



The formulae for the case in which the ring is circular, has no 

 rotation round its axis, and is not acted on by applied forces, 

 though, of course, easily deduced from the general equations 

 (14), (13), (6), are more readily got by direct application of first 

 principles. Let P be such a point in the axis of the ring, and 

 C, A, B such constants that J (Co> 2 + Aw 2 + Bz; 2 ) is the kinetic 

 energy due to rotational velocity <o round D, any diameter 

 through P, and translational velocities u along the axis and v 

 perpendicular to it. The impulse of this motion, together with 

 the supposed cyclic motion, is therefore compounded of 



momentum in lines through vS^ u + l alo "S *« « is > . 



[_Bv perpendicular to axis, 



and moment of momentum Co> round the diameter D. 



Hence if OX be the axis of resultant momentum, (cc, y) the 

 coordinates of P relatively to fixed axes OX, OY; the inclina- 

 tion of the axis of the ring to ; and £ the constant value of the 

 resultant momentum, we have 



fcos0=Aw + I; — fsin0=Bvj fy=Co>; "] 

 and ' ["-(IB) 



i=Mcos#— -vsin 0; i/ = u sin 0-\-v cos 0; = co.j 



Hence for we have the differential equation 



A€^+?hsm0+^Z*m2e]=O, . (16) 



which shows that the ring oscillates rotationally according to the 

 law of a horizontal magnetic needle carrying a bar of soft iron 

 rigidly attached to it parallel to its magnetic axis. 



When is and remains infinitely small, 0, y, and y are each 

 infinitely small, x remains infinitely nearly constant, and the 

 ring experiences an oscillatory motion in period 



9 / BC 



^V [I+(A-B)tf](I- ^ Ai?) , 



compounded of translation along OY and rotation round the 

 diameter I). This result is curiously comparable with the well- 

 known gyroscopic vibrations. 



