230 Dynamics. 



Jt being the ratio of the circumference to the diameter. There- 

 fore, 



that is, the time t' is independent of the height h' from which 

 the body sets out. 



Of the Moment of Inertia. 



- 



Fig.167. 354. Let M. M 7 , M'', be any masses without gravity, and let 

 them be considered as points situated in the same plane with the 

 point F, and connected together, and with the point J 1 , in such a 

 manner as not to admit of any change in their relative distances, 

 or of any motion except about the point F, or about an axis pass- 

 ing through jF, perpendicularly to the plane in which they are sit- 

 uated. Let us suppose that these masses receive at the same 

 time impulses according to the lines w, w', z/, directed in the 

 above plane, and such, that if the masses were free, they would 

 have velocities represented by these lines respectively, it is pro- 

 posed to determine the motion that would ensue. 



We decompose, according to the principle of D'Alembert, the 

 133. velocities w , w', w", each into two others, one of which shall be 

 effective, and the other such, that if the masses M, M', M", had res- 

 pectively only this velocity, they would remain in equilibrium. 



Now it is manifest that the velocities, which the bodies are 

 supposed to have, since they admit only of a rotation about the 

 point F, must be perpendicular to the radii R, R', R''. Moreover, 

 in order that these velocities may take place, that is, not mutually 

 disturb each other, it is necessary that they should be proportion- 

 al to these radii, or to the distances respectively from F. Accord- 

 ingly, the communicated velocities to, a/, a>", being decomposed 

 into 1 the effective velocities t>, i/, T", and the velocities M, w', w", 

 with which the masses would be in equilibrium about the point F, 



