140 ON THE MOTION OF SOLIDS THROUGH A LIQUID. [CHAP. V. 



It is plain that f , TJ O f , X , i , V Q) are the components of the 

 impulse of the cyclic fluid motion which remains when the solid is 

 (by forces applied to it alone) brought to rest. 



122. As a simple example we may take the case treated by 

 Thomson *; viz. where the solid is a circular ring (of any form of 

 section), and has therefore only one aperture. If we take the axis 

 of the ring as axis of a?, we see by the same reasoning as in Art. 

 116 that if the situation of the origin in this axis be properly 

 chosen we may write 



+ KK*. 



Hence % = Au + i; Q , r) = Bv, = Bw, 



X = Pp, p = Qq, v = fir. 



The fourth of equations (20) then gives ~ 0, or p const, as is 



obviously the case. Let us suppose that p 0, and that the ring 

 is slightly disturbed from a state of steady motion parallel to its 

 axis. In the beginning of the disturbed motion v, iv, q, r are small 

 quantities whose squares and products we may neglect. The first 



of (20) then gives -j- = 0, or u = const., and the remaining equations 

 become 



Eliminating r, we find 



= -(Au + ^{(A-B}u+^}v ......... (48). 



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

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



* Phil Mag. Nov. 1871. 



