VOL. LXX.] PHILOSOPHICAL TRANSACTIONS. 733 



&c. respectively; then from the same principles the effect of all the forces on a : 

 the eftect on b :: — + — + — + &cc. : — -\ [- — -f- &c. which quantities 



AD AE ' AF BD ■ BE ' Bl' ' 



put equal to p and q respectively, and then - : - :: A»ra : bw :: ac : bc; whence it 

 appears that, (putting gc + ga = ac and gc — gb = bc) the distance gc = 



A X Q X AG — B X p X B . ^ rpi^^ Same conclusion might have been deduced from 



Bxp — Axy . ° 



this consideration ; that if any number of forces act on a lever, the effect on any 



point of that lever is just the same as if a force, equivalent to the sum of these 



forces, had acted at their common centre of gravity; find therefore their common 



centre of gravity, and conceive a force equivalent to them all to be communicated 



to that point, and the problem is reduced to the case of the first proposition. 



If any of the forces had acted on the opposite side of the lever, such forces must 



have been considered as negative. 



If there be any number of bodies placed on the lever, and a single force act 

 at D, it will appear from the same principles that the point c, about which they 

 begin to revolve, will be the point of suspension to the centre of percussion d; 

 and the same conclusion will be obtained, if the bodies be not situated in a 

 straight line. As a direct investigation however, is always to be preferred to 

 conclusions drawn from induction, it may be thought proper, before we apply 

 any of the foregoing principles to the case of the action of bodies on each other 

 by impact, to show how such a direct investigation, to determine the point 

 about which a body, having a motion communicated to it, begins to revolve, 

 may be obtained; previous to which however, some further considerations are 

 necessary. 



Prop. Q. If a force act on a body in any given direction, not passing through 

 the centre of gravity; to determine the plane of rotation, and the direction in 

 which the centre of gravity begins to move, with its motion after. — Conceive a 

 plane aybz (fig. 0) to be supported on a line ab passing through its centre of 

 gravity g ; and suppose a force to act at any point d in that line, and in a direc- 

 tion perpendicular to the plane; then it is manifest, that such a force can give 

 the plane no rotatory motion about ab. Imagine now the support to be taken 

 away while the force is acting at d ; then it is evident, that as the plane had no 

 tendency to move about ab as an axis, and the taking away of the support can 

 give it no such motion, it will, by cor. 1, prop. 8, begin its progressive motion 

 in the direction in which the force acts; and as the force is supposed not to act 

 at the centre of gravity, it must at the same time have a rotatory motion about 

 some axis, which, as it has no motion about ab, must lie somewhere in the 

 plane, and perpendicular to ab; and consequently in ipso motiVs initio the plane 

 of rotation must be perpendicular to the plane aybz. Let lcm, perpendicular 

 to AB, be the axis about which the plane begins to revolve, and /;, q, be two equal 



