196 MOTION OF SOLIDS THROUGH A LIQUID. [CHAP. VI 



131. As a simple example we may take the case of an annular 

 solid of revolution. If the axis of x coincide with that of the ring, 

 we see by reasoning of the same kind as in Art. 123, 4 that if 

 the situation of the origin on this axis be properly chosen we 

 may write 



_l_ /.- .-\ ...2 /I \ 



~T~\ K > K ) K v 1 ;- 



Hence f = J.w + f > r) = Bv, =Bw,} 



A, = Pp, p = Qq, v=Qr j " 



Substituting in the equations of Art. 117, we find dp/dt=Q, or 

 p = const., as is obviously the case. Let us suppose that the ring 

 is slightly disturbed from a state of motion in which v, w, p, q, r 

 are zero, i.e. a state of steady motion parallel to the axis. In 

 the beginning of the disturbed motion v, w, p, q, r will be small 

 quantities whose products we may neglect. The first of the 

 equations referred to then gives du/dt = 0, or u = const., and the 

 remaining equations become 



Q^= [( A-B)u + S}, "' (3) ' 

 Eliminating r, we find 



Exactly the same equation is satisfied by w. It is therefore 

 necessary and sufficient for stability that the coefficient of v on the 

 right-hand side of (4) should be negative ; and the time of a small 

 oscillation, in the case of disturbed stable motion, is 



BQ 



We may also notice another case of steady motion of the ring, viz. where 

 the impulse reduces to a couple about a diameter. It is easily seen that the 

 equations of motion are satisfied by , rj, {, X, /n = 0, and v constant ; in which 

 case 



u = - A r = const. 



Sir W. Thomson, 1. c. ante p. 176. 



