ON THE 



MOTlOiNS OF FREELY SUSPENDED AND GYROSCOPIC PENDULUMS, AND 



ON THE PENDULUM AND GYROSCOPE AS EXIllBmNG 



THE ROTATION OF THE EARTH. 



Let the point of suspension of the penduhim be taken as the origin of rectangu- 

 lar co-ordinates, the axis of z vertical (downwards), and those of x and y in the 

 plane of the horizon, the former directed to the east, tlic latter to the north. 



The forces which act on the pendulum (considered as concentrated in its centre 

 of oscillation) are 



1st. Gravity, the resultant of the earth's attraction, and of the centrifugal force 

 of its rotation. 



2d. The tension of the pendulum cord. 



3d. The force of inertia, the components of which are represented by the differ- 

 ential coefficient ---^- -'■{ i-^-; and 

 at' at' at' 



4th. Tlie disturbing forces arising from the earth's rotation. 



5th. The resistance of the air. 



If we represent the length of the string by /. its tension by N, and the force of 

 gi-avity by g, and neglect the forces named in the 4th and 5th categories, we shall 

 have the three equations : 



fVx Nx 



~de~ T 



(Vz Nz , 



-9- 



df' I 



The forces due to the earth's rotation which disturb the relative motions of a 

 projectile, or any material particle moving near the eartli's surface, have been 

 expressed by Poisson (Journal de I'Ecole Polytechnique Cahier, 26) as follows : — ' 



' These expressions are perfecth' jreneral and applicable to all problems Tvhich involve relative 

 motions near the earth's surface, whether of solids or fluids. They (or their equivalent) appear in 

 the work of Laplace (Mec. Cel., Vol. IT,), as well as of Poisson (from whom I have quoted them), 

 in the investi<rations of the motions of projectiles. It is somewhat extraonliiiaiy that l)oth of these 

 great analysts should have failed to perceive their remarkable application in the motions of the 

 gyroscope and pendulum, to the exhibition of the earth's rotation, although the latter has seemed to 

 desire such an exhibition in the sentence: "Quoique la rotation de la torre soil niaintenant etablie 



(15) 



