MECHANICS. 



37 



Fig. 6T. 



Hence the tension of the second rope is 

 twice that of the first : in like manner, 

 the tension of the third rope is twice 

 that of the second, and so on, the 

 weight being equal to twice the tension 

 of the last rope. If then, as in the pre- 

 sent instance, there be four ropes, the 

 tension of the first is P, that of the 

 second 2 P, that of the third 2 x 2 P or 

 4 P, that of the fourth 2x2x2Por8 

 P, and therefore the weight W will be 

 16 P. 



It is obvious that each rope, which is 

 added to such a system, will double its 

 effect, and that the condition of equili- 

 brium expressed mathematically will be 



W=2"P; 



n expressing the number of distinct 

 ropes. 



In this case the effect of the weights 

 of the pullies themselves is neglected ; 

 but it is evident that they act against 

 the power, and therefore that it requires 

 a certain power to sustain them, even 

 when unloaded. 



(84.) The power of this system may 

 be greatly increased without increasing 

 the number of ropes, by substituting 

 fixed pulleys for the hooks to which 

 each rope is finally attached. This 

 method was applied to a single move- 



able pulley in fig. 58, and fig. 68 ex- 

 hibits a series of such pulleys. By 

 Jig. 58, it follows that the tension of the 

 second rope is three times that of the 

 first, which is equal to the power. The 

 tension of the third is three times that 

 of the second, or nine times that of the 



Fig.68. 



first, and so on, the weight being three 

 times the tension of the last rope. 



In the present example there are but 

 four ropes, and the weight is 81 times 

 the power. The relation between the 

 power and weight in such a system 

 mav be expressed mathematically thus : 



W=3"P; 

 n being the number of distinct ropes. 



The effect of the weights of the pul- 

 leys themselves in this case is also in 

 opposition to the power, and robs the 

 machine of part of its efficacy. 



(85.) A powerful system is repre- 

 sented in/#. 69. In this system each 

 rope is finally attached to the weight. 

 The first rope sustains a part of the 

 weight which is equal to the power. 

 The tension of the second rope is twice 

 that of the first, and therefore it sustains 

 a part of the weight equal to twice the 



