ON THE MOTIONS OF CONNECTED BODIES. 67 



one sense indeed the remark is true ; thus one man can do no more by a 

 powerful machine in ten hours, than ten men can do by a weaker machine 

 in one hour ; but in other senses the assertion is often erroneous ; for by 

 increasing the mechanical advantage to a given degree we may in some 

 cases considerably increase the performance of a machine without adding 

 to the force. 



According to the nature of the force employed, and to the construction 

 of the machine, a different calculation may be required for finding the best 

 proportions of the forces to be employed ; but a few simple instances will 

 serve to show the nature of the determination. Thus, in order that a 

 smaller weight may raise a greater to a given vertical height, in the 

 shortest time possible, by means of an inclined plane, the length of the plane 

 must be to its height as twice the greater weight to the smaller,* so that the 

 acting force may be twice as great as that which is simply required for 

 the equilibrium. This may be shown experimentally, by causing three 

 equal weights, supported on wheels, to ascend at the same time as many 

 inclined planes of the same height but of different lengths, by means of 

 the descent of three other equal weights, connected with the former three 

 by threads passing over pullies. The length of one of the planes is twice 

 its height, that of another considerably more, and that of a third less : if 

 the weights begin to rise at the same time, the first will arrive at the top 

 before either of the others. (Plate V. Fig. 76.) 



If a given weight, or any equivalent force, be employed to raise another 

 equal weight by means of levers, wheels, pullies, or any similar powers, 

 the greatest effect will be produced if the acting weight be capable of sus- 

 taining in equilibrium a weight about twice and a half as great as itself. 

 This proposition may be very satisfactorily illustrated by an experiment. 

 Three double pullies being placed, independently of each other, on an axis, 

 round which they move freely, the diameters of the two cylindrical por- 

 tions which compose the first being in the ratio of 3 to 2, those of the 

 second as 5 to 2, and those of the third as 4 to 1, six equal weights are 

 attached to them in pairs, so that three may be raised by the descent of the 

 other three, on the principle of the wheel and axis. If then we hold the 

 lower weights by means of threads or otherwise, and let them go, so that 

 they may begin to rise at the same instant, it will appear evidently that 

 the middle pulley raises its w r eight the fastest ; and consequently, that in 

 this case, the ratio of 5 to 2 is more advantageous than either a much less 

 or a much greater ratio. If the weight to be raised were very great in pro- 

 portion to the descending weight, the arrangement ought to be such that this 

 weight might retain in equilibrium a weight about twice as great as that 

 which is actually to be raised. If the descending weight were a hundred 

 times as great as the ascending weight, the greatest velocity would be 

 obtained in this case, by making the descending weight capable of holding 

 in equilibrium a weight one ninth as great as itself. (Plate VI. Fig. 77.) 



The proportion required for the greatest effect is somewhat different, 



when the heights through which both the weights are to move are limited, 



as they usually must be in practical cases. Here, if we suppose the opera- 



* Whewell's Dyn. c. 4, 4. 



F2 



