PROFESSOR MOSELEY ON THE THEORY OF MACHINES. 
303 
Whence* we obtain for the modulus of the pulley, 
U 
■ = {' +T + ^ cos ' sin ^} U 2 + {T 
+ 
W p (cos g 3 + cos < 2 3 ) sin </> 
2 a cos * 
}S,. . (24.) 
If both the strings be inclined at equal angles to the vertical, on opposite sides of 
it, or if < 13 = < 23 = t, so that cos q 3 + cos ; 23 = 2 cos /, the modulus becomes 
U l = {> +f + ¥ cos ' sin ^} U 2 + {^ + ^ sin ?’} S i (25-) 
If one part of the cord passing over a pulley have a horizontal, and the 
other a vertical direction, as, for instance, when it passes into the shaft of a 
mine over the sheaf or wheel which overhangs its mouth, then one of the angles 
h. 3 ,' 2.3 (equation 24.) becomes , and the other 0 or •r, according as the tension 
of the vertical part of the cord is upwards or downwards, so that cos q 3 -f- cos / 2 3 
= + 1, the sign + being taken according as the tension on the vertical branch of 
the cord is upwards or downwards: moreover in this case 1 = j, and cos t = -y- } 
therefore by equation (24.), 
u i = { 1 +£ + £ 7 ? *'*} u . + t{ d ±7I 
sin 
}s,. 
( 26 .) 
If the two parts of the cord passing over the pulley be parallel, and 
their common inclination to the vertical be represented by /, so that q 3 = / 23 
= i ; then, since in this case L = 2 a, we have by equation (23.), neglecting 
• • E p 
terms of more than one dimension in — and — , 
a a ’ 
u i - { 1 + a + ~i sin ?} U 2 + T { 1 + (a + T ') e sin p} ’ • • ( 27 ‘) 
77 
in which equation, i is to be taken greater or less than and therefore the sign of 
cos t is to be taken positively or negatively, according as the tensions on the cords 
act downwards or upwards. If the tensions are vertical, / = 0 or -r, according as they 
act upwards or downwards, so that cos t = + 1. If the parallel tensions are hori- 
zonfal, then / = ^ , and the terms involving cos 1 in the above equations vanish. 
If both parts of the cord passing over a pulley be in the same horizontal straight 
line, so that the pulley sustains no pressure resulting from the tension of the cord, but 
only bears its weight , then • — and the term involving cos / in equation (25.) vanishes. 
It is, however, to be observed, that the weight bearing upon the axis of 
the pulley, is the weight of the pulley increased by the weight of the cord 
which it is made to support ; so that if the length of cord supported 
by the pulley be represented by s, and the weight of each unit of length by g, then is 
* See Section 6, Equation 10. 
