157 
1914-15.] The Theory of the Gyroscope. 
The “rising” of a top due to excentric friction at the pivot is also 
readily accounted for by the action of the deviating force. 
/ 
4. In an experiment devised by Foucault, the axis OC of the gyroscope 
is restrained to move in a horizontal plane, but is otherwise free. The 
earth’s angular velocity (co) may be resolved into co sin X about the vertical 
and o) cos X about the N. and S. line, where X is the latitude. If (p be the 
angle which OC makes to the E. of N., the latter component gives to C 
a velocity o) cos X sin (p downwards, and so calls into play a deviating force 
Cncio cos X sin (p in the horizontal plane, tending to diminish (p. Meantime 
the N. and S. line is rotating about the vertical with the angular velocity 
(iOsinX. We have then 
A— - o)t sin X) = - CncD cos X sin <p 
(18) 
or 
A(jf) + Cno) cos X sin </> = 0 
(19) 
where, as is easily seen, A must be taken to include the moment of inertia 
of the frame of the gyroscope about the vertical through O. For a small 
oscillation about the N. and S. direction the period is 
277 ^ (- 20 ) 
In the “ gyrostatic compass ” the frame of the gyroscope is suspended 
by a wire, or floats on mercury, so that the axis OC has two degrees of 
freedom, horizontal and vertical. The theory is therefore modified. If 0 
be the inclination of OC below the horizontal plane, which we will suppose 
small, the vertical velocity of C is cos X sin 0 + d, and we have therefore, 
for the horizontal motion, 
A^ = - Cw(to COS X sin (^ + C • • • • (21) 
Again, owing to the horizontal motion <p, there is a component of deviating 
force tending to increase 0, so that 
-p^BO + Cncp ( 22 ) 
where B is the moment of inertia of the apparatus about a horizontal axis 
at right angles to OC, and the term —p^BO represents the “restoring 
