The Rev. E. R. Anstice on the Motion of a Free Pendulum, 379 

 These equations may be also written as follows : — 



, ,^p „ { -kV cos m't+Jdm! cos m"t} . . . (6.) 

 -vfr= ^,. /\, ■■ { — A;W' cos m't + k!'m! cos m'V} + 



^^^i^^{-m"sinm'^ + m'sinm"^} (7.) 



The principal object of the problem appears to be the determina- 

 tion of 6 when % is a maximum for a given number of vibrations. 

 For maximum values of % we shall have 



*S + -^^f =«• ...•■••• (80 



The preceding equations give all the circumstances of the motion 

 for small oscillations^ but I have not yet attempted their solution 

 in any particular case. 



Newcastle-upon-Tyne, 

 September 5, 1851. 



LVII. On the Motion of a Free Pendulum. 

 % Me Rev. R. R. Anstice/ ilf.^.* 



I. \ PLANE is rigidly connected with an axis, which axis 



■^^ rotates with an uniform angular velocity = b, carrying 

 the plane along with it. A material particle is constrained to 

 move in the said plane^ and also acted upon by a central attract- 

 ive force varying directly as the distance^ and situated in the 

 intersection of the axis and plane. To determine the motion, v 



This I shall afterwards prove will be the same as the small 

 oscillations of a simple pendulum at the earth's surface, free to 

 move in any azimuth : b will then be the angular velocity of the 

 earth's rotation. The axis will correspond in direction with that 

 of the earth, and the plane with the horizontal plane at the place 

 of observation. 



Refer the motion to three axes mutually at right angles. 

 Take origin at intersection of axis of motion and plane ; make 

 axis of motion axis of z. Then the plane of o^y will correspond 

 in direction with the earth's equator. 



Let If m, n be the cosines of inclination to the axes of oc, y 

 and z of the normal to the rotating plane at any time t. Then 

 n will be constant and = sine latitude. / and m will be func- 

 * Communicated by the Author, 



