MECHANICS. . 137 



been free to move by itself, to tbe velocity which 

 it has as a part of the system. 



This is proved, by considering that A (w - v) is the 



i A i , J$b v C cv D d v , 



motion lost bv A, and > ? , the 



a a a 



motions gained by B, C, and D ; and that these 

 last must, by D'ALEMBERT'S principle (120.), be 

 in equilibria with the former ; whence the equation 



Aa(u v) = (Bb* + Cc* + T> f P) -, from which 



a 



the value of v is obtained. 



Prom the demonstration, it is evident, that the mo- 

 mentum of the system relatively to S, is found by 

 multiplying each body into the square of its dis- 

 tance from S, and the sum into the velocity at A. 



The angular velocity is found by dividing both side* 

 by a ; this gives 



v __ A au 



a ~~ A 2 + B b 2 + C c 2 + D d* ' 



The quantity is the value of the angle described 



round S in one second, expressed in parts of the ra- 

 dius. If it is required in degrees and minutes, the 

 above value must be multiplied by 57.29578, the 

 number of degrees ip an arch equal to the radius. 



These propositions hold not only of a system of bodies 

 in one plane, but of a system in different planes, if 

 we suppose perpendiculars drawn from them to the 

 axis of rotation. 



S17. If 



