The Rev. A. Thacker on the Motion of a Free Pendulum. 277 



Eliminating X^ Y, Z, the equations (2.) become o^oiiaH 



d^x U w ^ - , . . ^. dz 



-—— -\ • Y — ^wcosXcos (ft)r smX) . -7- 



dt^ m I ^ ' dt 



— co^ cos^Xcos {cot sinX) {^ cos {cot sin X) +'y sin (cot sin X) } 



— 0)^ sin X cos X sin {cot sin X}.{a—z)=0 



d^y . R ?/ _, ^ • , . ■ ^sdz 



_^ ^ . ^-^2cocosXsin(cotsmX) -rr- 



dt^ m I ^ ^ dt , . 



^^ -w^cos^Xsin (ft)«f sinX){^cos (w^siiix) +y sinpVsinXJXj,, 



— co^ sin X cos X cos {cot sin X) . (« — 5') = 



J/2 "^ m' 1 "-^ "^ ^^^ I ^^^ \^^ ^^^ ) • ^ "^ ^^^-^ (^^si'^^) • "^ ^ 



— o>^ sin X cos X{^ sin {cot sin X) —y cos (o)^ sin X) } 

 + ft)2cos''^X.(«~-j)=0. 



These equations hold for any value of co. In the case we are 



considering co is small^ namely — — — — — - ; the terms multi- 



X<v P\ \j\J /\ \j\j 



plied by cd and co- are small and periodical j and if these be 

 neglected^ we have ^^ Y\ ^ /ma 



(3.) 



df^ ^m'l 



^^2 '^ m' I 



\W ... (4.) 



7\j : / sn> 



d^z ^ z 



which are equations of exactly the same form as mose'wliicli 

 apply to the motion of a pendulum suspended from a fixed point 

 in space ; the motion^ therefore^ is the same with regard to the 

 revolving axes as it would be with regard to fixed axes, if the 

 earth had no rotation. The angular velocity of the horizontal 

 axes being co sin X, it follows that the orbit will appear to revolve 

 at that rate round the vertical. 



I am. Gentlemen, *.'..>! s/. 



Your obedient Servant^ 

 Trinity College, Cambridge, . A. Thacker. 



June 1, 1851. xmV rrr^ / 



[It is much to be desired that the approximation should be carried 

 on one step further, and that at least the general effect should be 

 made out of such of the neglected terms in the above equations as 

 contain the first power of w. If the oscillations arenas considerable 



