MECHANICS. 



power. In like manner the third sus- 

 tains a part equal to four times the 

 power, and so on, the part sustained by 

 each rope being double that which is 

 sustained by the preceding one. 

 i In this case the weight is equal to 

 the sum of the tensions of all the strings, 

 and in the example represented m.fig. 

 69, it is thirty- one times the power. 



Fig. 69. 



It is not difficult to give a general 

 mathematical investigation of the rela- 

 tion between the power and weight. 

 The tensions of the several ropes by 

 which the weight is sustained are P, 

 2 P, 2 2 P, 2 3 P, &c. and if n be the 

 number of ropes, the tension of the last 

 is2 n ~ l P. So that we have 



W=P. (l+2 + 2i+23+: 2"-') 



But the series within the parentheses 

 being in geometrical progression*, its 

 sum is 2 n 1. Hence we have 



W= (2" 1)P. 



(86.) Such a system may be ren- 

 dered much more powerful without 

 increasing the number of ropes, by 

 passing each rope round a pulley, and 

 finally attaching it to the moveable 

 pulley over which it first passed, as in 

 Jiff. 70. The numbers placed upon the 

 several ropes express the parts of the 



* Sec DABLSV'S Popular Alycbra, p. 97 



weight which they respectively support, 

 the power being the unit In the system 

 in the figure the weight is 188 times the 

 power. 



The general mathematical investi- 

 gation of the relation between the 

 power and weight is similar to that in 

 (85.) The parts of the weight sup- 

 ported by each successive rope is 2 P, 

 6 P, 18 P, 54 P, &c. and that which i 

 supported by the last rope, the number 

 being n t is 2. S."- 1 P. Hence we have 

 AV = 2P(1 +3 + 32+3"+ ....3"-,).. 

 Summing the geometric series within 

 the parentheses, we have 



W=(3"-l.)P- 



(87.) The various systems of pullieS 

 which we have here described are not 

 offered to the student as machines 

 which he will frequently find adopted 

 in practice, but rather as an exercise n 

 the combination of these engines, and 

 as the best means of impressing upon 



