732 PHILOSOPHICAL TRANSACTIONS. [aNNO IJSO. 



point about which the bodies begin to revolve; for, considering the lever to be 

 produced to c, that point must have moved over a space cc equal to go, when 

 the lever is come into the position uivb: draw co perpendicular to cb, and go 

 perpendicular to civ, then o will be the centre of rotation at the commencement 

 of the motion. For conceive co to be a lever; then the lever abc has a circular 

 motion about c, while that point is moving from c to c, and consequently the 

 point o is carried forward in a direction parallel to cc by this motion; but as 

 the lever co is carried by a circular motion about c in a contrary direction, it is 

 evident that that point of the lever co must be at rest where these two motions 

 are equal, as they are in contrary directions. Now the velocity of c in the direc- 

 tion cc: velocity of g about c :: Gq : eg :: (by sim. triang.) co : cg, and the ve- 

 locity of the point g about c : velocity of the point o about c :: cg : co; hence 

 ex aequo the velocity of c in the direction of cc, or of o in the direction op 

 parallel to cc, is equal to the velocity of the same point o in a contrary direc- 

 tion arising from its rotation about c, and consequently o, being a point at rest 

 must be the centre of rotation in ipso mot6s initio. Also, because ma is equal 

 and parallel to nb, ab must be equal and parallel to mn, therefore the angular 

 velocity is just the same as if the force fh had not acted. The centre o of rota- 

 tion at the beginning of the motion being thus determined, every thing relative 

 to the motion of the bodies, after they are at liberty to move freely, may be de- 

 termined as in the preceding propositions. 



Cor. 1. Hence it appears, that whatever be the magnitude or direction of the 

 force communicating the motion, or the point at which it acts, the centre of 

 gravity will move in a line parallel to the direction of the force; for the triangles 

 FHD, Gqtu being similar, gw must be parallel to fd. 



Cor. 1. The same is manifestly true for any number of bodies. For let e 

 (fig. 5) be a 3d body, and conceive it to be connected with the other 2 bodies a 

 and B in their centre of gravity g: then if fd represents the force acting at the 

 point D, it is evident from the last corol. and the "Zd prop, that the centre of gra- 

 vity moves with the same velocity and in the same direction, as if the same mo- 

 tion had been communicated at g in a line rg parallel to fd, and that the centre 

 of gravity has the same velocity communicated to it, as if the 1 bodies had been 

 placed at g : conceive therefore the bodies a and b to be placed at g, and let the 

 force act at d, and then from the last corol. the centre of gravity g, of the 3 

 bodies, will move in a line parallel to the direction of the force communicated. 

 In the same manner it may be proved for any number of bodies. 



Scholium. — The method here made use of to determine the point of rotation 

 in ipso motfis initio, when a single force acts at any point d, may be applied, 

 when any number of forces act at different points at the same time. For let 

 «, |3j y, &c. (fig. 1) represent the forces acting on tlie lever at the points d, e, f. 



