1887.] in an Infinite Liquid. 55 
When the motion is steady 
G— hw--G — const. = vy, 
v = Co, = const. = CO, 
E=7=XK=p=—0. 
In order to obtain the disturbed motion, we must have recourse 
to Kirchhoff’s equations of motion*; we shall also suppose that 
the co-ordinate axes are fixed in the ring. 
Putting for shortness 
R ’ 
the equations of disturbed motion are, 
Pu — POQv + yo, = 0, 
Py — yo, + POu = 0, 
Aw, + Zy+ (C— A) Oo, = 0, 
Aw, — Zu — (C—A) Qe, =0. 
Whence, if p be the period of oscillation, 
L=y+ 
Pp —PO 0 Y 
EOL Ep —¥ 0 , 
0 Z Ap (C—A)Q 
-Z 0 —(C—A)Q Ap 
or 
A®P'p!+ P[2ZAy + {(C—A)*+ A} PO?) p?-+ (P(C —A) 02+ Zy}?= 0. 
The two values of p” given by this equation are both real and 
negative. Hence the values of p are imaginary, and therefore the 
motion is stable. 
If O = 0, the roots are 
Ly 
p=tta/ ap 
and are imaginary provided Ry+P¢,>Pry. Hence, if A> P the 
motion, when there is no rotation, is always stable; but if P> & it 
may be unstable. 
[If a prolate solid be moving parallel to its axis and there is 
no circulation, the motion will be unstable unless the angular 
velocity about its axis is sufficiently great; and the preceding 
article shows that when there is circulation but no angular velocity, 
* Lamb, Motion of Fluids, p. 126. 
