MECHANICS. 



in motion. Hence the proportion be- 

 tween the weight, friction, and pressure 

 continues to be the same ; a double or 

 treble pressure always produces a double 

 or treble degree of friction, and so on. 



[The constant proportion which is 

 found to subsist between the friction 

 and the pressure may be expressed in 

 reference to the angle, ABE, which 

 gives a force down the plane equal to 

 the friction. Let this angle be called 

 X, let F be the friction, and P the pres- 

 sure. By what has been already proved, 

 we have 



F;P :: AErBE 

 but AE ;BE ::tan.X; 1 



/. F : P : : tan. x : i 



.'. F=Ptan. X. 



Thus it appears that the tangent of 

 this angle always expresses the ratio of 

 the friction to the pressure.] 



What we have already stated as to 

 the independence of the friction on the 

 magnitude of the surface of contact, 

 may also be established experimentally 

 by the inclined plane. For on whatever 

 side the body W, is placed upon the plane, 

 the angle X, wDl be found to be the 

 same ; and therefore the proportion of 

 the friction to the pressure will be the 

 same. 



(9.) Another law deduced from ex- 

 periment is, that " friction is an uni- 

 formly retarding force." This is a law 

 respecting which no difference of opi- 

 nion whatever subsists ; and the results 

 of all experiments which have been in- 

 stituted on the subject are in perfect 

 accordance. 



It will be recollected, than an uni- 

 formly accelerating force, as explained 

 in our first treatise on Mechanics, is 

 one which produces an increase of ve- 

 locity in the moving body, which is pro- 

 portional to the time of its motion; 

 and the motion of a body excited by 

 such a force is characterized by several 

 remarkable properties, such as " that 

 the spaces described from the beginning 

 of motion are proportional to the squares 

 of the times of describing them ; the 

 spaces described in equal successive 

 intervals are as the odd numbers," &c. 

 Now an uniformly retarding force, on 

 the other hand, is one which destroys a 

 portion of the velocity of the moving 

 body ; and the quantity thus destroyed is 

 proportional to the time of the motion. 



If a body be urged at the same time 

 by two forces, the greater an uni- 

 formly accelerating force, and the lesser 

 an uniformly retarding force, it is evi- 



dent that the effect will be, that the 

 body will move with an uniformly ac- 

 celerating force which is equal to the 

 difference of the two forces to which it 

 is subjected. But if the force which 

 retards the body be not uniform, while 

 the force which accelerates it is so, then 

 the difference of these forces, with which 

 the body will move, will not be an uni- 

 formly accelerating force, since the 

 want of uniformity in the retarding 

 force will plainly affect the difference of 

 the two forces. If, therefore, a body 

 move with an uniformly accelerated 

 motion, being urged by two forces, an 

 accelerating and a retarding force, we 

 are warranted in concluding, that if the 

 accelerating force be uniform, the re- 

 tarding force must also be uniform, for 

 otherwise, according to what has just 

 been explained, the motion of the body 

 would not be uniformly accelerated. 



These observations being premised, 

 we are now prepared to explain the 

 experiments by which friction is proved 

 to be an uniformly retarding force. 



(10.) An apparatus such as has been 

 described in (7), is provided, the hori- 

 zontal plane, AB, being of considerable 

 length. The body, C D, is placed near 

 the extremity, A, and a weight is sus- 

 pended at S, sufficient to move the 

 body along the plane from A towards 

 B. The descent of the weight, S, is 

 measured by a graduated vertical scale, 

 similar to that used in Atwood's ma- 

 chine (First Treatise, p. 13), and the 

 rate noted by a clock in exactly the 

 same manner. In moving along the 

 plane from A to B, the body, C D, is 

 affected by two forces, one of which, 

 viz. the force with which the weight, S, 

 would draw it, independently of fric- 

 tion, is an uniformly accelerating force ; 

 the other, is the retarding force arising 

 from the friction. According to what 

 we have proved (9), it will immediately 

 follow, that the friction is an uniformly 

 retarding force, if we can shew by ex- 

 periment that the motion of the body is 

 uniformly accelerated. For since the 

 entire urging force of the weight, S, is 

 an uniformly accelerating force, and the 

 motion which actually obtains is also 

 uniformly accelerated, the retarding 

 force must be uniform (9). Let the body, 

 C D, commence its motion with a beat 

 of the clock, and let a stage be adjusted 

 by successive trials upon the vertical 

 scale, so that the weight, S, will strike it 

 with the second beat. The space through 

 which C D will move in one second will 



