138 CELESTIAL MECHANICS. I. ii. 11. 



oscillation will be in a vertical plane : we shall then have 



of half the greatest angle that the thread forms with the 

 vertical line, and its square half the verse sine of that 



angle. The time of the oscillation will then be T—tts/- 



!'*(l)..^+(t!)M^')'+(SI)--(=i^)--i'- 



Corollary 2. If the oscillation is verv small, -—- , 



2r 



being a very minute fraction, may be neglected in compa- 

 rison with unity : we may therefore call, in this case, T— 



V 



fts/ — , and we may consider the small vibrations as iso- 



9 

 chronous, whatever their comparative extent may be. 



Corollary 3. AYe may, therefore, employ expe- 

 riments on the length of a pendulum, vibrating in a given 

 time, for the determination of the variations of the inten- 

 sity of gravitation in different parts of the earth. If z be 

 the height through which a body would fall in the time T, 



we have z—^g T^ (232); consequently since T^zutt^ — , 



9 

 Z-zz^TT^r ; hence we may determine the space described by 



a gravitating body with the greatest precision by means of 



the pendulum. 



Scholium. It has been found, by very accurate expe- 

 riments, first made by Newton, that the length of the pen- 

 dulum vibrating in a given time is the same, whatever is 

 the nature of the substances composing it : whence it fol- 

 lows, that gravitation acts equally on all bodies, producing 

 in them the same velocity in the same time ; that is, in the 

 absence of a resisting medium. 



