PROFESSOR MOSELEY ON THE THEORY OF MACHINES. 303 



Whence* we obtain for the modulus of the pulley, 



TT r, , E 2p . 1 ,, . f I^ , Wp(co8i,„ + C08io«)8in6') ^ 



^■ = t'+V + T<">^'""^/U^ + |-+ 'I.eoB. }S|- ■ (24.) 



If both the strings be inclined at equal angles to the vertical, on opposite sides of 

 it, or if /j^^ = <2;j = ', so that cos /j 3 -f cos /gj = 2 cos /, the modulus becomes 



Ui = {l +- + ^cosisin^jU2 + |^ + ^^sin^jSi (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 



/i3,i23 (equation 24.) becomes g, and the other or t, according as the tension 



of the vertical part of the cord is upwards or downwards, so that cos t^^ + cos /gj 



= i Ij the sign + being taken according as the tension on the vertical branch of 



the cord is upwards or downwards : moreover in this case ' = 7, and cos / = J-, 

 therefore by equation (24.), 



Ui = {l+T + '4-'^'«^}u2 + i{D±^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 t^^ = '23 



= t ; then, since in this case L= 2 a, we have by equation (23.), neglecting 



E p 

 terms of more than one dimension in — and — , 



TT fi . E ,2p . 1 _, . D r , , /2 . WcosA . l , , 



Ui = |l+T+TSi^^j^2 + -7{i + (a +-D-)fS^^^|- • (27.) 



in which equation, / is to be taken greater or less than -^j and therefore the sign of 



cos / is to be taken positively or negatively, according as the tensions on the cords 

 act downwards or upwards. If the tensions are vertical, / = or t, according as they 

 act upwards or downwards, so that cos < = 4; 1. If the parallel tensions are hori- 



If 

 zontaly then < = o ' ^^^ *^® terms involving cos / 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 < = 2 * ^^^ 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 /;/», then is 



* See Section 6, Equation 10. 



