41 



motion of the axis to become for a time sensibly one of uniform pro- 

 gression ; it then becomes oscillatory again, the amplitudes of the 

 oscillations being smaller than before. 



THEOREM II. If the outer ring be fixed in any position so as to 

 restrict the axis of the gyroscope to a fixed plane, the motion of the 

 axis, when a weight is attached as above, is the same whether the 

 instrument be set rotating or not. It is proved that the angular 

 motion of the axis is determined by an equation of the same form as 

 that of a circular pendulum, which does not involve the angular 

 velocity of rotation impressed on the gyroscope. 



THEOREM III. If the gyroscope be set rotating rapidly, and its 

 axis of figure be constrained, as in Theorem II., to move very freely 

 in a plane fixed with regard to the horizon, the axis will tend to take 

 the position of the projection on the given plane of the line drawn 

 through the centre of gravity of the gyroscope, parallel to the axis 

 of the earth, in such a way that the earth and the gyroscope may 

 turn in the same direction ; while, if the axis be perfectly free, it will 

 move exactly in the same way as the axis of a telescope directed con- 

 stantly towards the same fixed star, their initial positions being sup- 

 posed parallel, as established experimentally by M. Leon Foucault 

 (Comptes Rendus, September, 1852). 



To prove this theorem, the angular velocity of the earth round its 

 axis is resolved into an equal and codirectional motion of rotation 

 round the line through the centre of gravity of the gyroscope parallel 

 to the earth's axis, and a motion of translation, the direction of which 

 is constantly changing, common to all parts of the earth. Of these 

 motions the latter is communicated to the gyroscope by the friction 

 of its base, and does not modify its position with regard to the 

 horizon. The first alone requires to be considered. In order to 

 estimate its effect, a rotation equal to it and round the same axis, but 

 in an opposite direction, must be supposed to be communicated both 

 to the earth and the gyroscope. This does not affect their relative 

 motion, and simplifies the problem, as it enables us to consider the 

 earth at rest. The relative motion of the gyroscope may therefore 

 be found by adding to the three components, round its principal 

 axis, of its instantaneous angular velocity of rotation, as found from 

 its equations of absolute motion, the components of this introduced 

 velocity of rotation, the moment of resistance of the given plane 



