VOL. LXX.] PHILOSOPHICAL TRANSACTIONS. 735 



passing through the centre of gravity g, perpendicularly in d ; then by cor. 2, 

 prop. 8, the centre of gravity g will begin its motion in a line parallel to pa, or 

 perpendicular to ln ; and consequently the centre c, about which the body be- 

 gins to revolve, must lie somewhere in the line ln. Now the centrifugal force 

 of any particle J!>, is/) X pc; let fall pa perpendicular to ln, then the effect of 

 that force at c, in a direction perpendicular to ln, will be /> X pa, and in the 

 direction cl it will he p X ca; but as the sum of all the quantities p X pa =: O, 

 and the sum of all the quantities p X ca = the body multiplied into cg, it 

 follows, from the same reasoning as in prop 3, that the point g will continue to 

 move in a direction perpendicular to ln, and also, as the forces /j X ca act in a 

 direction perpendicular to that in which the centre of gravity moves, its motion 

 must be continued uniform. In the following propositions therefore, we sup- 

 pose the axis of the body, after the commencement of the motion, to continue 

 perpendicular to the plane passing through the direction of the force, and the 

 centre of gravity of the body, and that the body itself is orthographically pro- 

 jected on that plane ; also in the case of the action of two bodies on each other, 

 the plane passing through the direction of the striking body and point of per- 

 cussion is supposed to pass through the centres of gravity of each body ; that 

 the axis of each body, after it is struck, continues perpendicular to that plane, 

 and that each body is reduced to it in the manner above described. 



Prop. 10. To determine the point about ivhich a body , ivhen struck, begins to 

 revolve. — Let lmno (fig. 7) represent the body, g the centre of gravity, and pa 

 the direction of the force acting at a, which produce till it meets ln, passing 

 through g, perpendicularly in the point d; draw pb perpendicular to pc, on 

 which, produced if necessary, let fall the perpendicular v>w; c being supposed 

 the point about which the body begins to revolve, and which, from the last prop, 

 is somewhere in the line ln. Because the body, in consequence of the force 

 acting at d, begins to revolve about c, and consequently if, immediately after the 

 beginning of the motion, a force were applied at n equal to it, and in a con- 

 trary direction, the motion of the body would be destroyed, it is evident, that 

 the efficacy of the body revolving about c, to turn the body about d, should 

 any obstacle be opposed to its motion at that point, must be equal to nothing; 

 for were it not, the body, when stopped at d, would still have a rotatoi-y motion 

 about that point, and consequently two equal and opposite forces applied at d 

 would not destroy each other's effects, which would be absurd. Now the force 

 of a particle />, in the direction pw, being p X pc, its efficacy to turn the body 

 about the point d, is p X pc X Tiw; but by sim. triang. d?(' : Bb :: ac -.pc, there- 

 fore Div = , and consequently the efficacy to turn tiie body about d = p 



X Tib X ac = p X ca X {dc — cb) = p X ca X BC — p X pc^; hence the 

 sum of all the p X ca X dc — the sum of all the p X pc" = O, and conse- 



