ON THE 



MOTIONS OF FRKKLY SCSl'KNDI.D AM) GYROSCOPIC PENDULUMS, AND 



ON THE 1'KMH U M AM) GYROSCOPE AS EXHIBITING 



THE ROTATION OF THE EARTH. 



l.i r the point of suspension of the pendulum be taken as the origin of rectangu- 

 l;ir co-ordinate-, the axis of 2 vertical (downwards), and those of z and y in the 

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



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

 of oscillation) are 



1st. (ira\it\, the resultant of the earth's attraction, and of the centrifugal force 

 of its rotation. 



V.M. The tension of the pendulum cord. 



:}d. The force of inertia, the components of which are represented by the diffcr- 



~ . , il'.r (Pll d*Z 



initial coefficient __; and 



ar (I I' at 



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

 ")th. The resistance of the air. 



If we represent the length of the string by 7. its tension by JV, and the force of 

 gravity by '/. and neglect the forces named in the 4th and 5th categories, we shall 



have the three equations: 



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

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

 expressed by Poisson (Journal de 1'Ecole Polytechnique Cahier, 26) as follows : ' 



1 These expressions are perfectly ffeneral and applicable to all problems which involve relative 

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

 the work of Lnplnre (Mec. Ccl, Vol. IV.), as well as of Poisson (from whom I have quoted them), 

 in the investigations of the motions of projectiles. It is somewhat extraordinary that both of those 

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

 pyr. .scope nnd 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 tcrre soil maintenant etablie 



(15) 



