42 LECTURE VI. 



imaginary body to three given directions, and collecting all the results into 

 three sums, which will represent the motion of the centre of inertia 

 [gravity] of the system. 



We have already presupposed this proposition, when we have employed 

 material 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 those of 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 

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

 The centre of inertia [gravity] of a system of bodies moving without dis- 

 turbance, is, therefore, 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 [gravity] from the same plane. And the 

 proposition will be equally true, if, instead of the shortest distances, we 

 substitute the distances from the same plane, measured obliquely, in any 

 directions always parallel to each other. This property is peculiarly appli- 

 cable to the consideration of the centre of gravity [weight], and affords also 

 the readiest means of determining its place in bodies of complicated forms. 

 (Plate III. Fig. 32.) 



We have already seen that the place of the centre of inertia [gravity] of 

 two bodies is not affected by any reciprocal action between them ; and the 

 same is true of the actions of a system of three or more bodies. We might 

 easily apply our experiment on the reciprocal action of two bodies to a 

 greater number, but we should throw no further light on the subject, and 

 the mode of obtaining the conclusion would be somewhat complicated. 



All the forces in nature, with which we are acquainted, act reciprocally 

 between different masses of matter, so that any two bodies repelling or at- 

 tracting each other, are made to recede or approach with equal momenta. 

 This circumstance is generally expressed by the third law of motion, that 

 action and reaction are equal. There would be something peculiar, and 

 almost inconceivable, in a force which could affect unequally the similar 

 particles of matter ; or in the particles themselves, if they could be pos- 

 sessed of such different degrees of mobility as to be equally moveable with 

 respect to one force, and unequally with respect to another. For instance, 

 a magnet and a piece of iron, each weighing a pound, will remain in equi- 

 librium when their weights are opposed to each other by means of a 

 balance ; they will be separated with equal velocities, if impelled by the 

 unbending of a spring placed between them, and it is difficult to conceive 

 that they should approach each other with unequal velocities in consequence 

 of magnetic attraction, or of any other natural force. The reciprocality of 

 force is therefore a necessary law in the mathematical consideration of 

 mechanics, and it is also perfectly warranted by experience. The contrary 



