1887.] 



Mechanism regarding Trains of Pulleys, Sfc. 



137 



In the particular case where r is a multiple of b, or r = nh, 



_ 2(a + l) +*(«*+ ■!) 

 0g * ~ 2*0 + 1) 



and if w = 1, 



l + O + l) 2 



log a 



2a (a; + 1) 



This equation gives a = 2'21 nearly, or practically a ratio of 11 to 

 5 for the diameters of the large and small drums in such a train as 

 has been indicated. 



Although it is manifest that the volume of a single pair of pulleys 

 with the same velocity ratio as this train of five pairs would be con- 

 siderably greater, it may be interesting to make the comparison. If 

 R' be the radius of the large pulley in the single pair, and as before 

 r of the small pulley, then E/ = ur t and the volume of the pair 

 V = 7rt(u 2 -\-l) r 2 . As before, the volume of the train is V = 

 7m£(> 2 + l)r 2 . 



Hence 



V % 2 + l X* n + 1 



If a? = 1*9 and n = 5, we shall have V'/V = 26*64, or the volume of 

 a single pair would be more than twenty- six times the volume of a 

 train of five pairs with the same velocity ratio. 



Another solution can be easily found if a train of drums were so 

 constructed that the volume of the spokes supporting the hoop of 

 each drum would be half the volume of a complete c^isk, in this 

 case 



V = 7n7rf[2(R + r)5 + J(R 2 +r 2 )] 



= tttH [2(tf+l)- + i(* 3 + l)]^, 

 L v y r /J iogx 



and if we make h = r, this gives, from dV/dx = 0, 



+ 2)2 + 1 



log X = 



2x(x + 2) ' 



which is satisfied by making x = 2*55, or the diameters of the large 

 drums would be to those of the small drums in the ratio of 51 to 20 

 in a train of least weight of drums such as here described. 



If in this case b in all the drums instead of being equal to r was 

 equal to the grea.ter radius H, we would have evidently 



V= m7rr 2 ^[2(x + l)x + l(a 2 + l)] 



log" u 



= l7rr 2 /[5x 2 + 4r+l] 



log X 



