Mr. A. H. Curtis on the Gyroscope. 397 



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 Foucauit 

 (Comptes Rendus, September, 1852). 



To prove this theorem, the angular velocity of the earth round it3 

 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 

 being taken into account in forming the equations of motion, and its 

 intensity supposed such as to counteract that part of the total angular 

 velocity of the axis which is perpendicular to the given plane. The 

 equation which determines the motion of the axis is shown to be 

 identical with that of a circular pendulum, and the motion con- 

 sequently one of oscillation, the mean position of the axis being that 

 in which it approaches, as close as the conditions of the question 

 permit, to the line drawn through its centre of gravity parallel to 

 the earth's axis, and in which it rotates in a direction similar to that 

 of the earth's rotation. Similar reasoning establishes the second 

 part of the theorem, which is theoretically true whether the gyro- 

 scope be set rotating or not. This result is, however, in practice 

 modified by the effects of friction ; but when a rapid rotatory motion 

 has been impressed on the gyroscope, it acquires a stability which 

 enables it to overcome to a great extent these effects. 



December 13. — Major-General Sabine, Treasurer and Vice-President, 

 in the Chair. 



The following communication was read : — 



"On the Surface-condensation of Steam." By J. P. Joule, 

 LL.D., F.R.S. 



In the author's experiments steam was passed into a tube, to the 

 outside of which a stream of water was applied, by passing it along 



