284 Dynamics. 



397. When a machine is already constructed^ the velocity 

 of its impelled and working points are determined ; and therefore 

 in order to obtain from it its maximum effect, we must seek for 

 the best proportion between the power and the resistance, as 

 these are the only circumstances over which we have any con- 

 trol, without altering the machinery. 



In order to find the ratio of p to r, which would produce a 

 maximum effect, it is requisite only to make r variable in the 

 formula above given ; but it often happens, that when r varies, 

 the mass n suffers a considerable change, although there are 

 other cases when the change in n is too inconsiderable to be 

 noticed. 



Let us, therefore, first take the case when r alone varies with- 

 out inducing a change in n. In this case, the expression for the 

 work performed, namely, 



rp D tf r 2 32 rfd 2 



m D 2 + n d 2 + i d 2 

 will be a maximum when 



tf 



as will be readily found by differentiating, putting the differential 

 equal to zero, and deducing the value of r. But according to 

 the experiments of Coulomb, the friction is in general equal to 

 of the resisting pressure. Hence we may omit /d, and 



1 fi 

 consider the resistance as = r + T l j r - r. Consequently, 



I O 



r = J^, and r M^j) X ^j. But if we consider the 

 fraction as so near 1 , that the substitution of the latter will 



lb 



not greatly affect the result, we shall obtain, by making p = 1 

 and D = 1 ? r = r ; that is, the resistance should be nearly one 



half of the force which would keep the impelling power in equi- 

 librium, a rule which is applicable to many cases where the 

 matter moved by the working point of the machine is inconsider- 

 able. 



