304 
PROFESSOR MOSELEY ON THE THEORY OF MACHINES. 
the weight sustained by the axis of each pulley represented by W -f- g s. Substitu- 
ting' this value for W and assuming cos / = 0 in equation (25.), we have for the modulus 
of the pulley in this case, 
Ui = ( 1 + — ) U 2 +-{D + (W + g s) § sin <p j- Sj (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 bend 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 U 1 represent the work done upon the cord, through the space S l5 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 ; U 2 after it has passed over the second, 
&c. ; and \J n after it has passed over the wth pulley or sheaf ; then 
Ui= (l + ~) U 2 -f — |d + (W + g s) § sin <p j- S x ; 
U g = (l + — ) U 3 + — D + (W + g s) § sin <p j- S l5 &c. See. ; 
U »=( 1+ v) U »-i + 4{ D + (W + /‘*)fsin?}s l . 
Eliminating the n — 1 quantities U 2 U 3 . . . U n i between these n equations, and 
E D p 
neglecting terms involving powers of —•> —■> — sin <p above the first, we have 
U,= 0 + vH + ir {D + OV + Mfsinf.Js. 
(29.) 
Let us now suppose that the rope, after passing horizontally over n equal pulleys, 
the radius of each of which is represented by a, 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 v that of its axis by g 13 and 
its weight by W x ; 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 U 2 represent the work done upon the rope through the space Sj after it has 
assumed the vertical direction or passed into the shaft, and let JJ n 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.), 
U. 
= {' 
+ - + 
«i 
Pi 
V2 
sin <p 
|u 2 + — |d+ ^ysin <p |s r 
