304 PROFESSOR MOSELEY ON THE THEORY OF MACHINES. 



the weight sustained by the axis of each pulley represented by W + i^ *• Substitu- 

 ting this value for W and assuming cos / = in equation (23.), we have for the modulus 

 of the pulley in this case, 



Ui=(l+f)u2 + i{D + (W + ^^)fsin^jSi (28.) 



In which equation it is supposed that although the direction of the rope on either side 

 of each pulley is so nearly horizontal that cos / may be considered evanescent, yet the 

 rope does so far hend itself over each pulley, as that its surface may adapt itself to the 

 curved surface of the pulley, and thereby produce the whole of that resistance which 

 is due to the rigidity of the cord. 



Let it now be supposed that there is a system of n equal pulleys, or sheaves of 

 the same dimensions, placed at equal distances in the same horizontal straight line, 

 and sustaining each the same length s of rope. 



Let Ui represent the work done upon the cord, through the space S^, by the moving 

 power, or before it has passed over the first pulley of the series ; Uj the work done 

 upon it after it has passed over the first pulley ; Ug after it has passed over the second, 

 &c.; and U„ after it has passed over the nth pulley or sheaf; then 



Ui=(l+|)u2 + -i-{D+(W + ^*)?6in?}s,; 

 U2=(l+v)u3 + -l{D + (W + f<.*)?sm?jSi,&c.&c.; 



U-.= (l +t) U.-1 + t{i' + (W + /**)?sm?}s.. 



Eliminating the w — 1 quantities U2 U3 . . . U„ j between these n equations, and 



E D o 

 neglecting terms involving powers of — ? — ^ — sin (p above the first, we have 



Ui = (l + v)u» + ^{D + (W + ^.)fsinf}s, (29.) 



Let us now suppose that the rope, after passing horizontally over n equal pulleys, 

 the radius of each of which is represented by «, and its weight by W, as in the pre- 

 ceding case, assumes at length a vertical direction, passing over a pulley or sheaf of 

 different dimensions, whose radius is represented by a^, that of its axis by f^, and 

 its weight by Wj ; as for instance, when the rope of a mine descends into the shaft 

 after having traversed the space between it and the engine, supported upon pulleys. 



Let U2 represent the work done upon the rope through the space S^ after it has 

 assumed the vertical direction or passed into the shaft, and let U„ represent, as before, 

 the work done upon it after it has passed over the n horizontal pulleys, and before it 

 passes over that which overhangs the shaft. Then by equation (26.), 



