504 MR. 8S. S. HOUGH ON THE OSCILLATIONS OF A 
Hence, the motion of the fluid will consist very approximately of a rotation, as if 
rigid, with angular velocity # about the line whose direction-cosines are 

Tee ¢ sin (\ — ot + €), 7 p cos (X — wt +), 1. 
This axis will itself describe a cone in period 27/Ko, and it will be so situated that 
the axis Oz lies in the plane containing the axes of revolution of the fluid and of the 
shell, and is between these two axes. The semi-vertical angles of the cones described 
by the axes of rotation of fluid and the shell will be in the ratio 3 : 2g + 1. 
The motion under discussion is that which would ensue if the shell were set 
rotating about its principal axis, while the fluid possessed a rotatory motion in the 
same period about some other axis. It is clear, that as €,, 6, and consequently E, 
diminish, the period of this oscillation will be prolonged ; that is to say, the motion of 
the axes of rotation will become slower. This motion will be reduced to zero when 
€,, €, vanish. In this case the internal surface of the shell is spherical, and the 
shell and fluid, of course, move independently. So far as this (apparent) oscillation 
is concerned, they will each be rotating with angular velocity w, but about 
different axes. 
From the expression for the ratio of the amplitudes, we see that when q is large, 
that is when the effective inertia of the fluid is large, compared with that of the shell, 
the disturbance of the shell will be considerable, compared with that of the fluid ; 
whereas if g be small, the disturbance of the shell bears to that of the fluid, a ratio 
which approximates to, but is always in excess of, 1 : 3. 
This oscillation has been previously examined by Hopkins (‘ Phil. Trans.,’ 1839) 
under certain special assumptions, as to the initial circumstances, and to the law of 
distribution of density in the shell. 
(c). Lastly, suppose \ = w(K, + Ge) (kK. + Ge) 
The approximate form of equations (46) is now 

yy. sgt EA sis B, 
8, {xy + geo} B— 16',B V/(«, + G61) (Ka + Ge) + Ce, * = 0, 
T 

y “ar / B, 
O's {mq + qq 5 A + OA V (1, + 961) (Ke + Gee) — Qa, a 0, 
eS 

B, Baty ase = : 
Bee (1 + 2e) + UV (1, + Ger) (HF Ye) = 0, 
>) Bs ¢ ea, Bs 
Lies (1 + 2) — iV (Kk, + Ge) (Kz ++ Ge) Sr 0. 
Hence approximately B,/7 = B,/7 = 0, and 
a; 16’, = ge", say ; 
J (ky, + Ge) == VJ (kz + Ge) 
