1887.] Mechanism regarding Trains of Pulleys, Sfc. 135 



circle, it may be understood as giving a rule for deducing the ratio of 

 tbe diameters of the wheels and pinions so that the sum of all their 

 circumferences shall be a minimum. Although economy of the cir- 

 cumferences of wheels, speed pulleys or drums in a train may not be 

 of much importance, is it not possible that economy of total weight 

 of material employed may be worthy of inquiry ? Reduction of weight 

 in the parts of a machine is not merely economy of materials em- 

 ployed in the structure, but in the case of moving parts it involves 

 economy of work by lessening the resistances due to friction. The 

 following problems have arisen from such considerations, and in all of 

 them, as well as in that studied by Young, if we call m the number of 

 similar pairs of wheels, speed pulleys or drums, C the circumference 

 of a large wheel, &c, and c of a small one in the same train, the 

 velocity ratio or value of the train u will be — 



u = (C/c) m = (R/r) m = x m , 



where x represents the ratio of the radii R and r of a large and a 

 small wheel or pulley. In all such problems we have therefore 



<m = log it/log x, 



and whether the question relates to the volume or circumference of 

 the wheels or pulleys the usual operations of the calculus will in 

 every case lead to a minimum. 



The volumes or circumferences of pairs of pulleys or wheels with 

 radii having the ratio x may in general be expressed in the form 

 Foj = a + bx + cx 2 , where a, b, and c are constants. On multiplying 

 this by m we have — 



V = 



losr x 



Hence 



1 dV F'x Fx 



logudx logx x(log x) 



1 d*V F"x 2F'x Fx 2Fx 



\ogu dx* logx x(logx)^ as 3 (log a?)" 3 x 2 (logx) 6 



F"x + ~Fx 2_ / F'x _ Fa? \ 



log x a? 3 (log x) 2 x log x \log x a? (log x) 2 J 



n , dV _ F'x Fx 



But as — — 0, = 0, 



ax logx x{\ogx) z 



, d*V log u /_ „ , Fx \ 



and — - = ( Fx"+ , 



ax* log x \ x" logx J 



