MECHANICS. 



35 



that part of the rope which extended 

 from the hook in the upper block to the 

 wheel a must pass over that wheel : this 

 foot of rope must evidently also pass 

 over the wheels b, c, and all the suc- 

 ceeding wheels. But the part of the 

 rope which extended from the wheel a 

 to the wheel b is also shortened by one 

 foot. This foot of rope must therefore 

 pass over the wheel b and all the suc- 

 ceeding wheels c, d, fyc. Hence one 

 foot of rope passes the wheel a in the 

 ascent, and two feet pass b. By con- 

 tinuing this reasoning, we shall find that 

 three feet of rope pass c, four pass c?, 

 and so on. Now the velocities with 

 which the wheels revolve (their diame- 

 ters being the same) is justly measured 

 by the quantity [of rope which passes 

 over them in the same time. Hence, 

 while the wheel a revolves once, b re- 

 volves twice, c three times, d four times, 

 and so on. Hence arises that inequa- 

 lity of wear which L we have already 

 mentioned. 



If we attempt to remove this defect 

 by fixing all the wheels on the same 

 pivots so as to compel them to turn with 

 the same velocity, we shall introduce 

 another source of friction and cause of 

 wear much greater than the former ; for 

 since the rope passes over the wheels 

 with different velocities, while they re- 

 volve with the same velocity, it must 

 necessarily scrape or slide more or less 

 on the grooves of all of them, one ex- 

 cepted. 



The great object, therefore, in the 

 construction of such a system of pullies 

 would be to make them all revolve on 

 then* axles in the same time, so as to 

 avoid unequal wear ; and yet that their 

 grooves or circumferences should have 

 different velocities equal to those of the 

 rope in passing over them. 



(80.) These ends were all attained 

 by a pulley invented by Mr. James 

 White. In order that the successive 

 wheels should revolve in the same time, 

 he constructed them of different magni- 

 tudes ; and so that their several cir- 

 cumferences would be equal to the 

 length of rope which passes over them 

 in the same time. This will be easily 

 understood by recurring to/^.62. Sup- 

 pose the circumference of the wheel a 

 to be one foot: it makes one revolu- 

 tion, while the lower block is raised 

 through one foot towards the upper 

 block. In this time three feet of rope 

 pass over the circumference of the 

 wheel c. If then the circumference of 



this wheel be three feet, it will revolve 

 once during the supposed ascent of the 

 weight. In like manner, the wheel e 

 will revolve once if its circumference be 

 five feet, and so on. - Thus, then, in 

 general it follows, that the several 

 wheels will revolve in the same time, if 

 their circumferences be as the numbers 

 1, 3, 5, &c. ; and in the same way it 

 may be proved, that the wheels in the 

 upper block will revolve in the same 

 time with each other, and with the 

 wheels in the lower block, if then- cir- 

 cumferences be as the numbers 2, 4, 

 6, &c., or, what is the same, as the suc- 

 cessive integers 1, 2, 3, &c. 



The circumferences of circles are 

 proportional to their diameters, and, 

 therefore, by constructing the several 

 wheels a, b, c, d y &c. with diameters pro- 

 portional to the successive integers 1 , 2, 

 3, &c., equality of wear would be ob- 

 tained. 



But still the multiplied friction of a 

 great number of different wheels would 



Fig. 63. 



