278 CELESTIAL MECHANICS, l.vil. 31. 



yii 



(y"* + s"2) Dwz=— 1^ C^W', and taking the fluxion, ^A. 



:z: ■ C, di being constant : and the body being sub- 

 ject only to the force of gravitation, P and Q will be =:0, 

 and jR will be constant; therefore d^T" z: dSfRydtDm '■=. 



SRydtDm ; =:SRyDmzz RSyDm = R cos6 Sy Dm 



-hi? sin 5 Sz^Dm : but since z" passes through the centre 

 of gravity of the body, we have Sy"Dmi=0 ; and if A be 

 the distance of the centre of gravity from the axis of mo- 

 tion y, we have Sz^Dmzzmh, m being the mass of the body, 



, d^" 7T^ . . 1 ddfl —mJiR sind 



whence --— iz mfiK sm 0. and -;— =: 7= . 



d^ ' dt^ C 



If we now consider a second body, of which all the atoms 



are united in a single point at the distance I from the axis 



x", we have in this case Czzmfl^, m' being the mass: and 



, , ^, dd9 — m'hR smO R . ^ rm 

 A=/; consequently -nr= 77s = -7- sm 6, Ihe 



two bodies will therefore have exactly the same oscillatory 

 motion, if their initial angular velocities, when their centres 

 of gravity are in the vertical line, are equal, and if / z: 



C 



— -. The second body here taken into consideration is the 

 mh 



simple pendulum, of which the oscillations have been de- 

 termined in § 11 (280) ; and we may always assign, by 

 means of this formula, the length / of the simple pendulum, 

 of which the oscillations are isochronous with that of the 

 solid here considered, which constitutes a compound pen- 

 dulum. It is thus that the length of the simple pendulum, 

 vibrating in a second, has been determined from observa- 

 tions on the vibrations of compound pendulums. 



