16 



MECHANICS. 



fall are proportional to the squares of 

 the times of the fall (29.), and the times 

 themselves are proportional to the velo- 

 cities (26.), it follows that the heights 

 are proportional to the squares of the 

 velocities. That a body may acquire 

 a double velocity, it is requisite that it 

 should fall from a fourfold height, and 

 so on. 



CHAPTER IV. On the Centre of 

 Gravity. 



(40.) WE have stated that, at a given 

 place upon the surface of the earth, the 

 force of gravity acts on all bodies in 

 lines which are parallel to each other, 

 and perpendicular to an horizontal, or 

 level plane. When it acts upon a single 

 body, it does not act, as it were, by a 

 single effort, but impresses a separate 

 force upon each particle of the body ; 

 and its total effect is composed of the 

 sum of all its effects thus produced 

 upon the particles. Now there is in the 

 body a certain point, at which, if the 

 attraction of gravity impressed a single 

 force, equal in intensity to the sum of all 

 its separate actions on the component 

 parts of the body, the ultimate effect 

 would be the same as it is under the 

 system of separate action which really 

 obtains. This point, the existence of 

 which we shall prove experimentally, 

 is called the centre of gravity. 



(41.) If the attraction of gravity 

 were confined in its action to one par- 

 ticular point, there are certain effects 

 which would very evidently ensue. 



First, if that point were supported or 

 fixed, the body would rest in any posi- 

 tion whatever in which it should be 

 placed. For the only cause which we 

 suppose to affect it so as to produce 

 motion, acts upon a point which we 

 suppose fixed. 



Secondly. If the body be perfectly 

 free to move, the point on which the 

 attraction acts will commence to move 

 in the direction of that attraction, and 

 in this case will, therefore, commence 

 to move in a line perpendicular to an 

 horizontal plane. 



Thirdly. If the body be suspended 

 by any point different from that at 

 which alone the attraction of gravity is 

 supposed to act, it will only remain at 

 rest in two positions, viz. when the at- 

 tracted point is immediately under or 

 immediately over the point of suspen- 

 sion. If the attracted point be in any 

 other position, the body will move round 



the point of suspension, all its parts 

 describing circles round that point, until 

 the attracted point settles directly under 

 the point of suspension. 



These effects will 

 be evident from a lit- 

 tle consideration. Let 

 A B be the body, and 

 P the point at which 

 it is suspended, and 

 round which it is ca- 

 pable of moving. Let 

 C be the point at 

 which the whole at- 

 traction of gravity is 

 supposed to act. 

 First, suppose this 

 point to be placed in 

 a line P D, vertically 

 under the fixed point 

 P. The attraction 

 then acting in the di- 

 rection of the line C D, will only pro- 

 duce a pull on the point C, which will 

 resist it, and no motion Jig. \ \ . 

 will ensue. 



Next, let the point 

 C be in a line verti- 

 cally above the fixed 

 point P. The whole at- 

 traction will now act in 

 the direction, C D, and I 

 will therefore produce a | 

 pressure on the point 

 P, which will be re- 

 sisted by that point, 

 and no motion wul en- 

 sue. 



Lastly, let C be in a 

 position neither direct- 

 ly above nor below the 

 fixed point P. Draw 

 C D' perpendicular to an horizontal 

 plane, and parallel to C D, and taking 



fig- 12. 



any portion C o from C, draw the 



