-j. = b SI sin <p r 2 , 



324 VII. DYNAMICS OF ROTATING BODIES. 



the integrals become 



250) 



If we put 



o + 1 sin tp t = ty, 

 the first becomes 



and the second, 

 252) ! 2 + r * - 2 H sin 9, - 



Introducing the above value of T 2 -^' putting a-\- 2b Rising? = c, and 

 neglecting ( 2 , 



This and 251) are the equations of the spherical pendulum, but we have 



co = iff 5i sin <p t f 



hence the axes of the ellipse described by the bob revolve around 

 the vertical with the angular velocity i&sinqp in the direction east- 

 south-west-north. This was verified by Foucault in his celebrated 

 experiment made in the Pantheon at Paris in 1852. 



1O6. Foucault's Gyroscope. Let us now consider the cele- 

 brated experiments by which, by means of a gyroscope, Foucault 

 demonstrated the rotation of the earth. Let us consider a symmetrical 

 gyroscope, suspended by its center of mass. If it is free to move, 

 and is started spinning about its axis of symmetry, it will evidently 

 by the principle of conservation of angular momentum, keep the axis 

 of angular momentum, which is here the axis of symmetry, pointing 

 in the same direction in space, so that this axis, while pointing 

 always at the same star, describes a circular cone with reference to 

 the earth. Instead of treating the general motion, which would lead 

 to too great complications, we shall treat two important cases, in 

 which the axis of symmetry is constrained to move either in a 

 vertical or horizontal plane. This we shall do by making use of 

 equations 29), 84, following the method of Hayward, who gave 

 those equations, in a paper in the Transactions of the Cambridge 

 Philosophical Society, Vol. 10, read in 1856. 



Suppose first the top constrained to move in a vertical plane, 

 and take for axis of Z, the axis of figure, which makes an angle # 

 with the earth's axis, for axes of X and Y axes fixed in the meridian 



