222 Dynamics. 



When the pendulum CB is drawn from its vertical position. 

 the force of gravity acting according to the vertical line AM is 

 not wholly employed in moving the body ; a part is exerted 

 against the point C. Let therefore the whole force of gravity, 

 represented by AM^ be decomposed into two others, represent- 

 ed the one by #JV, directed according to G^JV, which will be 

 destroyed, and the other by JlP which urges the body along the 

 arc JIB. Now as the radius CA is perpendicular to the arc, it 

 will be seen that the motion is here decomposed in the same 

 manner as in the case above considered, where the body is 

 supposed, without any material connection with C, to descend 

 along the arc AB, which has for its radius the length AC of the 

 pendulum. Accordingly every thing which we have said is ap- 

 plicable to pendulums. The following are some of the conse- 

 quences which are derived from the preceding investigation. 



345. We have found for the duration t of an oscillation, the 

 following expression, namely, 



Hence, for another pendulum whose length is a', and which is 

 urged by a different force of gravity, or one that is capable 

 of giving the velocity g' in a second, we shall have, by call- 

 ing if the duration of an oscillation in this second case, 



hence we derive the proportion, 



g' Vg 



that is. if two pendulums of different lengths are urged by different 

 gravities, the durations of the oscillations are as the square roots of 

 the lengths of the pendulums , divided by tht square roots of the quan- 

 tities which denote these gravities. 



346. As gravity is the same in the same place, we shall have 

 for pendulums of different lengths vibrating in the same place or 

 same part of the earth, g = #, and consequently in this case the 

 proportion becomes 



