54 LECTURE VI. 



mon centre of inertia ; determining the centre of inertia of this iniaginarj 

 body and tlie tliird 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, we shall be able to pro- 

 duce an experimental 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 of a system, 

 of moving bodies, by supposing their masses to be united into one body, and 

 tliis 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 imaginary body to 

 three given directions, and collecting all the results, into three sums, which 

 will represent the motion .of the centre of inertia of the s-ystenu 



We have already presupposed this proposition, when we have employed ma- 

 terial bodies of finite magnitude, that is, systems of material atoms, to represent 

 imaginary bodies of the same weight, condensed into their centres ; and it 

 now appears, that the velocity and direction of the motions of such bodies as 

 we have employed, agree precisely with thoseof our imaginary material points. 

 We cannot attempt to confirm this law by experiment, because the deductions 

 from the sensible consequences of an experiment would require nearly the 

 same processes as the mathematical demonstration. 



It is'obvious that the result of any number of uniform and rectilinear mo- 

 tions, thus collected, must also be a uniform and rectilinear motion. The 

 centre of inertia of a system of bodies moving without disturbance, is, there- 

 fore, either at rest, or moving equably in a right line. 



The mass, or weight, of each of any number of bodies, being multiplied by its 

 distance from a given plane, the products, collected into one sum, will be 

 equal to the whole weight of the system, multiplied by the distance of the 

 common centre of inertia from the same plane. And the proposition will be 



