CENTER of gravity of two or more bodies, a point so situated in a right line joining the centers of these bodies, that, if this point be suspended the bodies will equi ponderate and rest in any situation. In two equal bodies it is at equal distances from both : when the bodies are unequal, it is nearer to the greater body, in pro portion as it is greater than the other ; or the distances from the centers are in versely as the bodies. Let A (fig. 5,) be greater than B, join A B, upon which •ake the point C, so that CA:CB ::B: A, or that A X CA =B X CB; then is C the center of gravity of the bodies A and B. If the center of gravity of three bodies be required, first find C the center of gravity of A and B ; and supposing a body to be placed there equal to the sum of A and B, find G the center of gravity of it and then shall G be the center of gravity of the three bodies A, B, and D. In like manner the center of gravity of any number of bodies is determined.
The sum of the products that arise by multiplying the bodies by their respective distances, from a right line or plane given in position, is equal to the product of the sum of the bodies multiplied by the dis tance of the center of gravity from the same right line or plane, when all the bodies are on the same side of it : but when some of them are on the opposite side, their products, when multiplied by their respective distances from it, are to be considered as negative, or to be sub ducted. Let I L, (fig. 6.) be the right
line given in position, C the center of gravity of the bodies A and B ; A a, B b, C c, perpendiculars to I L in the points a, b, and c: then if the bodies A and B be on the same side of I L we shall find A + A a + bxBb=A + B X C c. For draw ing through C, the right line M N paral lel to I L meeting A a in M, and B b in N, we have A :B::B C: A C by the proper ty of the center of gravity, and conse quently A:B:: BN:A M, or A X AM butAxAa+BX13 6= AxCc+A xAM+Bx Cc—B xB N= A xCc+BX Cc =A+1.1x Cc. When B is on the other side of the right line I L (fig. 7,) and C on the same side withA,thenAXAa—BxBb=AX Cc+AxAM—B XBN+BXCc= A 4. II X C and when the sum of the products of the bodies on one side of 1 L, multiplied by their distances from it, is equal to the sum of the products of the bodies multiplied by their distances on the other side of 1 L, then C c vanishes, or the common center of gravity of all the bodies falls on the right line 1 L.
Hence it is demonstrable, that when any number of bodies move in right lines with uniform motions, their com mon center of gravity moves likewise in a right line with an uniform motion ; and that the sum of their motions, estimated in any given direction, is precisely the same as if all the bodies in one mass were car ried on with the direction and motion of their common center of gravity.