ON THE MOTIONS OF SIMPLE MASSES. 41 



ratio of the products, obtained by multiplying the weight of each body 

 by the number expressing its velocity ; and these products are called 

 the momenta of the bodies. We appear to have deduced this measure of 

 motion from the most unexceptionable arguments, and we shall have occa- 

 sion to apply the momentum thus estimated as a true measure of force ; at 

 the same time that we allow the practical importance of considering, in 

 many cases, the efficacy of forces, according to another criterion, when we 

 multiply the mass by the square of the velocity, in order to determine the 

 energy : yet the true quantity of motion, or momentum, of any body, is 

 always to be understood as the product of its mass into its velocity. Thus 

 a body weighing one pound, moving with the velocity of a hundred feet in 

 a second, has the same momentum and the same quantity of motion as a 

 body of ten pounds, moving at the rate of ten feet in a second. 



We may also demonstrate experimentally, by means of Mr. Atwood's 

 machine [Plate I. Fig. 11], that the same momentum is generated, in a 

 given time, by the same preponderating force, whatever may be the quan- 

 tity of matter moved. Thus, if the preponderating weight be one sixteenth 

 of the whole weight of the boxes, it will fall one foot in a second instead of 

 16, and a velocity of two feet will be acquired by the whole mass, instead 

 of a velocity of 32 feet, which the preponderating weight alone would have 

 acquired. And when we compare the centrifugal forces of bodies revolving 

 in the same time at different distances from the centre of motion, we find 

 that a greater quantity of matter compensates for a smaller force ; so that 

 two balls connected by a wire, with liberty to slide either way, will retain 

 each other in their respective situations when their common centre of 

 inertia [gravity] coincides with the centre of motion ; the centrifugal force 

 of each particle of the one being as much greater than that of an equal 

 particle of the other, as its weight or the number of the particles is smaller. 



But it is not enough to determine the centre of inertia [gravity] of two 

 bodies only, considered as single points ; since in general a much greater 

 number of points is concerned : we must therefore define the sense in which 

 the term is in this case to be applied. We proceed by considering the first 

 and second of three or more bodies, as a single body equal to both of them, 

 and placed in their common centre of inertia [gravity], ; determining the 

 centre of inertia [gravity] of this imaginary body and the third body, and 

 continuing a similar process for all the bodies of the system. And it 

 matters not with which of the bodies we begin the operation, for it may be 

 demonstrated that the point thus found will be the same by whatever steps 

 it be determined. When we come to consider the properties of the same 

 point as the centre of gravity [weight] we shall be able to produce an ex- 

 perimental proof of this assertion, since it will be found that there is only 

 one point in any system of bodies which possesses these properties. (Plate 

 III. Fig. 31.) 



We may always represent the motion of the centre of inertia [gravity] 

 of a system of moving bodies, by supposing their masses to be united into 

 one body, and this body to receive at once a momentum equal to that of 

 each body of the system, in a direction parallel to its motion. This may' 

 often be the most conveniently done by referring all the motions of this 



