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



Or 



C4-D C— D 



X= — 2 — sin (fl/-f €—</)) H ^ — siii(a/+e + <^) 



Y= — ^ — cos(fl/ + e— <^) ^ — cos(a/ + € + </)) 



By comparing equations (7.) and (8.), we get at once 

 C+D 



(8.) 



.CssA+B D = A~B 





} 



^-« .„. \ (^•) 



Therefore equations (7.) refer to an ellipse, whose axes are con- 

 stant, and =2(A + B), 2(A— B) respectively; but the direction 

 of which axes have an uniform angular motion of regression 

 (/. e. contrary to that of the earth), and which —hn— eartVs 

 angular velocity x sine latitude. If one of the two, A-f-B, 

 A— B=0. The motion in that case will be rectilinear. 



III. It now only remains to prove (what is in fact self-evident) 

 that the problem already discussed is that of the pendulum at 

 the earth^s sm-face free to move in any azimuth, provided the 

 oscillations thereof are small. 



Consider, then, the motion of a material particle acted on by 

 gravity, and constrained to move in a spherical surface attached 

 to the earth and rotating with it. 



Make the earth's axis the axis of z, and take origin at the 

 point where the vertical of the place of observation cuts the same. 



Let /, m, n be the cosines of inclination of the vertical to the 

 axes of Xy y and z ; .*. w is constant, and = sine latitude. 



Let R = distance of particle when at lowest point from origin ; 

 .*. R/, Rm, Rn will be coordinates of lowest point. 



Let R/ + X, Rwi -f- y, Rw + r be coordinates of particle at time t ; 

 r= radius of spherical surface; 



.-. (r/-a7)2 + (m-y)2+(r7i-2')2=r2 . . (1.) 



will be the equation to the surface ; 



, a? y z 



I , m— -, n 



r r r 



will be the cosines of inclination of the normal of said surface. 



Let g be the force of gravity at the given place ; /', m', w' the 

 cosines of inclination of the direction in which it acts. 



The normal accelerative force of reaction may be divided into 

 two ; one constant, the same as is exercised when there is no 

 osciUationj and the particle remains in (apparent) rest at its 



