228 



CENTRE OF GRAVITY. 



But for the effects of friction and atmospheric resistance, the body would 

 continue for ever to oscillate equally from side to side of the line P F. 



The phenomenon just developed is only an example of an extensive class. 

 Whenever the circumstances which restrain the body are of such a nature that 

 the centre of gravity is prevented from descending below a certain level, but 

 not, on the other hand, restrained from rising above it, the body will remain at 

 rest if the centre of gravity be placed at the lowest limit of its level ; any dis- 

 turbance will cause it to oscillate around this state, and it cannot return to a 

 state of rest until friction or some other cause have deprived it of the motion 

 communicated by the disturbing force. 



Under the circumstances which we have just described, the body could not 

 maintain itself in a state of rest in any position except that in which the centre 

 of gravity is, at the lowest point of the space in which it is free to move. This, 

 however, is not always the case. Suppose it were suspended by an inflex- 

 ible rod instead of a flexible string : the centre of gravity would then not only 

 be prevented from receding from the point of suspension, but also from ap- 

 proaching it ; in fact, it would be always kept at the same distance from it. 

 Thus, instead of being capable of moving anywhere within the sphere, it is 

 now capable of moving on its surface only. The reasoning used in the last 

 case may also be applied here, to prove that when the centre of gravity is on 

 either side of the perpendicular P F, it will fall toward P F, and oscillate, and 

 that, if it be placed in the line P F, it will remain in equilibrium. But in this 

 case there is another position, in which the centre of gravity may be placed 

 so as to produce equilibrium. If it be placed at the highest point of the sphere 

 in which it moves, the whole force on it will then be directed on the point of 

 suspension, perpendicularly downward, and will be entirely expended in pro- 

 ducing pressure on that point ; consequently the body will in this case be in 

 equilibrium. But this state of equilibrium is of a character very different from 

 that in which the centre of gravity was at the lowest part of the sphere. In 

 the present case, any displacement, however slight, of the centre of gravity, 

 will carry it to a lower level, and the force of gravity will then prevent its re- 

 turn to its former state, and will impel it downward until it attain the lowest 

 point of the sphere, and round that point it will oscillate. 



The two states of equilibrium which have been just noticed are called stable 

 and instable equilibrium. The character of the former is, that any disturbance 

 of the state produces oscillation about it; but any disturbance of the latter state 

 produces a total overthrow, and finally causes oscillation around the state of 

 stable equilibrium. 



Let A B, fig. 10, be an elliptical board resting on its edge on a horizontal 

 plane. In the position here represented, the extremity P of the lesser axis 



Fig. 10. 



