108 Statics. 



the same may be said of the other ropes ; whence 



IL = IM sin IML ; NQ = IM sin NPQ ; 

 accordingly, 



q i p n IM : IM sin ML -f /./V sin JVPQ + &c., 

 : : 1 : sin IML + sin NPQ + &c. 

 If the ropes are parallel and consequently vertical, the angles 

 IML, NPQ, &c., will be right angles, and their sines will be 

 each equal to radius, or 1. Therefore the power in this case 

 will be to the weight as 1 is to the sum of so many units as there 

 are ropes meeting at the moveable pulley. Hence it will be 

 seen, that if one of the extremities of the rope is attached to the fixed 

 Fig. 88, pulley, the power will be to the weight as unity is to double the num.' 

 ber of moveable pulleys ; and if the extremity of the rope is attached 

 g.J s< 89 ' to Hit moveable pulley, the power will be to the weight as unity is t& 

 double the number of moveable pulleys, plus 1. 



182. The general proposition above demonstrated, holds 

 true, whether the ropes are in the same plane or not ; and if the 

 obstacle to be overcome be not a weight, that is, if the direction 

 of the whole power of the pulley be not vertical, we have only to 

 substitute for the angles which the ropes are supposed to make with 

 the horizon, those which they would make with a plane perpen- 

 dicular to the whole action of the pulley. In figure 93, for exam- 

 ple, the power q is to the force exerted at G, as radius is to the 

 sum of the sines of the angles made by the several ropes (meet- 

 ing in CF) with a plane perpendicular to FG. 



183. If several sets of pulleys are employed, it will be easy 

 after what has been said to assign the ratio of die power to the 

 weight. In figure 93, for example, the ropes being supposed 

 parallel, the power q will be to the force exerted in the direction 



is*. CB, as 1 is to 5. Now this last force performs the office of a 

 power with respect to the system of pulleys BA, and accordingly 

 is to the weighty, as 1 to 4. Therefore the power q is to the 

 weight p, as 1 X 1 is to 4 X 5, that is, as 1 to 20. 



184. In all that precedes, we have- supposed the system of 

 pulleys destitute of gravity and friction, and the ropes perfectly 

 flexible. We shall see hereafter what allowance is to be made 

 for friction and the stiffness of the ropes. With respect to the 



