MECHANICS. 



thus be determined. By the same pro- 

 cess \ve may find the spaces described 

 in two, three, four seconds, or in as 

 great a number of seconds as the length 

 of the. plane, A B, will permit. If the.se 

 spaces be found to be in the same pro- 

 portion as the squares of the numbers 

 1, -2, 3, 4, &c., then the motion of the 

 body, CD, is uniformly accelerated, 

 and otherwise not. Treatise I.) In a 

 series of very accurately conducted ex- 

 periments, instituted by the late Pro- 

 fessor Vines of Cambridge, this law was 

 found to be observed with the utmost 

 exactness. 



Hence we infer, that " friction is an 

 uniformly retarding force." 



The same conclusion mi^ht be esta- 

 blished by experiments on the inclined 

 plane. If the plane be elevated to such 

 an height as to cause the body to de- 

 scend, it will be found that the descent 

 is uniformly accelerated. Since the 

 force down the inclined plane, indepen- 

 dent of friction, is an uniform force, it 

 follows, upon the same principle as 

 before, that friction must be an uni- 

 formly retarding force. 



(11*.) The law which we have ex- 

 plained of the proportionality of the 

 friction to the pressure under given cir- 

 cumstances, was derived from very ex- 

 tensive and varied experiments instituted 

 by several philosophers, but particularly 

 by Coulomb and Xime?ies ; nor was it 

 ever called in question until the late 

 Professor Vince of Cambridge renewed 

 the inquiry, and instituted experiments, 

 the results of which led him to conclude 

 that this law does not obtain, or at least 

 not accurately. We shall now explain 

 the manner in which Professor Vince 

 conducted the experiments from which 

 he deduced results differing from those 

 of Coulomb. 



When the body, C D, is moved along 

 the plane, A B, by the effect of the 

 weitrht, S, omitting the consideration of 

 the "friction, the accelerating force with 

 which it would move would depend on 

 the proportion of the weights of C D 

 and S. It follows, therefore, that if 

 C D and S be both increased in the 

 same proportion, the accelerating force, 

 independent of the friction, will remain 

 unchanged. If the friction be propor- 

 tional to the pressure, it will be in- 

 creased in the same proportion as the 

 weight of C D ; but then the weight 

 C D, which it retards, is proportionally 

 increased, and therefore the degree of 

 retardation which it produces must_ be 



the same. Hence it follows, that since, 

 by increasing the weights of C D and S 

 in the same proportion, the two forces 

 which affect C D, viz. the accelerating 

 and retarding force, remain unaltered, 

 their difference, which is the actual 

 accelerating force with which C D is 

 moved, will remain unaltered. Thus, 

 it follows that, granting the proportion- 

 ality of the friction to the pressure, no 

 change should be produced in the rate 

 of motion of C D, when both C D and 

 S are doubled, or trebled, or increased 

 or decreased in any other proportion. 



[A rigorous mathematical investiga- 

 tion of this may be satisfactory to some 

 readers. Let m be the quantity of matter 

 in C D, and ml that in S ; let g be the 

 accelerating force of gravity, and / the 

 constant ratio of the friction to the pres- 

 sure; id g is the moving force which 

 draws the combined masses m and m'. 

 Therefore, the accelerating force with 

 which they would be moved, indepen- 



ml s 

 dently of friction, would be ~^r/ since 



the accelerating force is equal to the 

 moving force divided by the quantity of 

 matter. But / m expresses a moving 

 force, which is equal to the friction of 

 C D with the plane ; and as this force 

 acts in retarding the combined masses 

 m and m', the corresponding retarding 



force is ^T^r- The actual accelerat- 

 ing force being the difference between 

 this and the former, is m _i_ ^ 

 This being put under the form 



+ 1 



it is evident that its value does not de- 

 pend on the absolute values of m and 

 m', but only on their ratio ; and that so 

 Ions: as that ratio remains the same, the 

 accelerating force with which C D 

 moves along the plane will remain un- 

 altered. 



If, upon experiment, it were found 

 that, by increasing the weights of C D 

 and S in the same proportion, the acce- 

 lerating force with which C D is moved 

 does not continue the same, but is in- 

 creased, what is to be inferred ? That 

 part of the accelerating force which is 

 independent of the friction, depends 

 entirely on the proportion of the weights 

 of C D and S, as has been already ex* 



