\,v\J; lx\\\.v5.i"'/i 



of a freelp'mp^ikdd'FMml^h:' ''^''^'^- 'Wn ^ 



in which -cr is the longitude of the apse measured from a fixed 

 line CU. JBoqcfwa ^enyiltj *j«)ii.t >/;>)/b : r ^amo 



From this equation Sve "can oBtaih the motion of thte ap^e 

 which results from any radial disturbing force. In order to 

 apply it to the case of the elliptic vibration of a freely suspended 

 pendulum^ we suppose the pendulum to be a point moving on a 

 sphere, and urged towards its lowest point by a force tangential 



to the surface = a sin r = — ^ =^. We may therefore sup- 



^ ^1+tanV ^ ^ 



pose that the motion takes place in a moveable spherical ellipse, 



the disturbing force being radial, acting outwards from the centre, 



n tan 7* 

 and equal to the difference of a tan r(l + tan^r) and ; =, 



Q vl+tan^r 



or to ~ ^ tan^ r, if the arc r be so small that all powers higher 

 than the third may be neglected. Substituting, therefore, for 

 — its value ^ tan^ r, and for cos^ r its approximate value unity, 

 we obtain 



(a^-/3"^)8t^= gtan^r ^!^ "^. "^ da>. . . . (9.) 



Let -i/r be an angle which satisfies the equations 



acos-xlr; 



i-, 2^06 



tan r 



acos'^/r; 



cosa)= — —J lis 06 



tan r 



and therefore 



smo); 



cos ft)' = ^, ^fff 



tanm 



smft)'= -, 



tanm 



The last equation may be reduced to 



E^= ^(l-{.2^^^^CO&2^|r + COB4^P^^d^|r. . (10.) 



Let Act- be the total variation in the value of ct, while the angle 

 'yjrj which varies uniformly with the time, increases from cipher 

 to any finite value, we have by integrating equation (10.), and 



substituting for u and /3 their approximate values j and-^, / being 



the length of the pendulum, - ' '^ '' ' '"'■ " ''" '^ ^ ' 



Sab / , a^ + b^ . J. *1 -y . vA /i-, ^ 



