730 PHILOSOPHICAL TRANSACTIONS. [aNNO l/SO. 



move in the direction of the rod, is (a -f b) X gc ; but from mechanics a X 

 AC + B X BC = (a + b) X GC : hence the centrifugal forces of the bodies A 

 and B give the centre of gravity a centrifugal force equivalent to its own centri- 

 fugal force, which, as the latter would cause that centre to move in the tangent 

 Gg, the lever not being fixed at c, it is manifest that the former will cause the 

 centre of gravity to continue its motion in the same direction. 



That this motion of the lever, in a direction from the centre c, is the only 

 motion which is communicated to it from the effect of the bodies a and b, is 

 manifest from hence : the bodies begin to revolve freely about the point c, and 

 consequently if the point c had been fixed, the bodies would have moved on with 

 a uniform angular velocity about c ; if therefore we suppose the lever not to be 

 fixed at c, as the efficacy of the centrifugal force which acts in the direction of 

 the lever is now suffered to take place, and no new external force is impressed on 

 either of the bodies, it is manifest, that if in the former case the bodies had no 

 efficacy to disturb the angular velocity of the lever, they cannot have any in the 

 latter ; consequently the angular velocity, and from what has been before proved, 

 the uniform motion of the centre of gravity in a right line, remain unaltered, 

 after the commencement of the motion. 



Prop. 5. In the time the bodies make one revolution, the centre of gravity will 

 move over a space equal to the circumference of a circle whose radius is cg, 

 {fig. 1.) — From the last prop, the angular velocity of the lever is continued uni- 

 form: hence the time of a revolution is just the same as if the point c were 

 fixed, and the bodies were to continue to revolve about that point as a centre, in 

 which case the centre of gravity g, in the time of a revolution, would evidently 

 describe the circumference of a circle whose radius is gc. This therefore is the 

 space the centre of gravity describes in a right line when the bodies move freely ; 

 for by the last prop, that centre is carried uniformly forward with the same 

 velocity. 



Cor. 1. Hence, if the quantity of the force acting at d vary, the velocity of 

 the centre of gravity will vary in the same ratio as the angular velocity. 



Cor. 1. Hence the point d may be found, where a force being applied, the 

 bodies shall make one revolution, while the centre of gravity moves over any 

 given space s : for let p ^ the periphery of a circle whose radius is unity, then 

 jb : I : : 5 : - = the radius of a circle whose circumference is the space to be passed 

 over in the time of a revolution, and which must therefore, by the prop, be 

 equal to go : the point c therefore being determined, d may be easily found ; 

 for from mechanics cg X dg is given ; and from cor. 3, prop. 1, when d comes 

 to A, c will coincide with B : cg X gd = ag X gb, and consequently dg = 



AG X GB 



CG 



