Pulley. 107 



Sometimes they are united upon the same axis as in figures 92, 

 93. The latter arrangement has the advantage of being more 

 compact ; but when a large number of pulleys are thus disposed 

 in the same block, the power being applied on one side instead 

 of being directed through the middle, the system is drawn awry, 

 and part of the force employed is lost by the oblique manner in 

 which it is exerted. This inconvenience does not belong to the 

 pulleys represented in figures 88, 89, 90, 91 5 and in that repre- 

 sented by figure 94, the peculiar advantages of the two systems 

 are united. Here two sets of pulleys having a common axis are 

 attached to the moveable block, and two to the fixed block, the 

 inner set in each case being of a less diameter than the outer 

 so as to allow a free motion to the rope. Then the rope com- 

 mencing at the middle of the upper block after being made to 

 pass over all the pulleys will terminate also in the middle. This 

 arrangement was invented by Smeaton. 



181. But whatever difference there may be in this respect in 

 the particular disposition of the pulleys, the ratio of the power to 

 the weight may always be found by the following rule. The 

 power is to the weight as radius, or sine of 90 , is to the sum of the 

 sines of the angles made by the several ropes (meeting at the moveable 

 pulley) with the Iwrizon. 



Indeed, if upon each of the ropes we take the equal parts 

 /,/, JVP, &c., to represent the tension, and upon each of these J^S- 88 ' 

 lines, as a diagonal, we form a parallelogram, having one pair of 

 its opposite sides vertical, and the other pair horizontal ; instead 

 of considering the weight p as sustained by the immediate ten- 

 sion of the rop^.s, we may regard it as supported by the hori- 

 zontal forces IK, JVO, &c., and the vertical forces /L, JVQ, &c. 

 Now the first being perpendicular to the action of the weight 

 contribute nothing to counterbalance this action; and in the 

 case of an equilibrium these horizontal forces mutually destroy 

 each other. The weight />, therefore, is wholly sustained by the 

 resultant, that is, by the sum of the vertical forces /L, JVQ, & c j 

 and the ropes being all equally stretched, it is evident that q is to 

 p as the tension of one of these ropes is to the entire sum of the 

 vertical forces. But in the right-angled triangles IML, JVQP, 

 &c., we have 



IM : IL : : 1 : sin IML ; NP or IM : NQ : : 1 : sin 



