158 Proceedings of the Eoyal Society of Edinburgh. [Sess. 
couple ” due to gravity and the tension of the wire, or to gravity and 
buoyancy, as the case may be.^ 
If we assume that 0, as well as 0, is small, and that both vary as 
we have 
( o C?2 (ocosA\ , iCn<T^ ^ X 
V '- A ) 
(23). 
Hence 
C?uo cos 
AB 
-2 = 0 
(24> 
Of the two values of o-^, one is greater than the greater, and the other 
is less than the smaller, of the two quantities and (C^io; cos X)/A. In 
practice n is very great compared with _p or co, and the two roots are 
cr2 = C%VAB and o-^ = ^ . . . (25) 
\^7l 
approximately. The former corresponds to a very rapid vibration, which 
is quickly checked by friction, and makes 
= (26) 
nearly. 
The second and more important root gives a slow oscillation in which 
n iCno- j iwcosX. 
^ = 9^ ^ 
ar 
(27> 
The period {^iTjcr) of this slow oscillation is in practice about 70 minutes,, 
and the ratio of 0 to 0 is therefore small. 
5. In the Schlick contrivance for steadying the rolling of a ship, a fly- 
wheel maintained in rapid rotation is carried by a frame which can swing 
about an axis at right angles to the medial plane of the vessel. The axis 
of the flywheel itself moves in this plane, and its standard position is 
upright, this being the position of stable equilibrium when the ship is at 
rest and there is no rotation, the frame being weighted with this object- 
The swinging of the frame about the transverse axis is resisted by frictional 
brakes. Briefly the principle of the contrivance is that the rolling of the 
ship produces a deviation of the axis of the flywheel in the medial plane,, 
and a consequent absorption of energy by friction, which means so much 
lost to the rolling vessel. 
If the angular displacements are small the equations of motion of the 
frame are obtained by a slight modification of the equations (14) relating ta 
* So that 27 t/^ is the period of oscillation when n - 0. 
