Rope Machine. 81 



the sequel, what allowance is to be made for the want of this 

 quality. Moreover, cords are first considered as destitute of 

 weight, regard being had afterward to their gravity. The 

 greater or less diameter of the cords also is not considered as 

 affecting the communication of forces; since we may always 

 substitute in imagination for these cords, considered as cylinders, 

 a line or thread answering to their axes, the force employed be- 

 ing considered as acting by means of this thread only. 



We employ cords to transmit the action of forces immediate- 

 ly, or in connexion with machines. But in order to judge of 

 the effects of powers applied to machines by means of cords, it 

 is necessary to ascertain the effects of which these powers are 

 capable, when they act by means of cords alone. 



141. Accordingly, let us consider three powers, p, q, r, as Fig. 56.. 

 acting the one against the other, by means of three cords Ap,Aq, 

 Ar, united at A by a knot; and supposing the directions A p, 

 A q, A r, to be known, we propose to determine the conditions 

 necessary to an equilibrium among these forces, and the ratio of 

 these forces. 



(1). It is evident, in the first place, that they must all three be 

 in the same plane, for if one, the force r, for example, were not in 

 the plane of the two others, we could always conceive it decom- 

 posed into two forces, one in this plane, and the other perpendic- 

 ular to this plane, and consequently perpendicular to each of the 

 two forces p, q ; this perpendicular force would not act, therefore, 40 - 

 in any way against the forces p, q ; and would accordingly have 

 nothing to be opposed to it, and an equilibrium with respect to it 

 could not take place. 



(2). These three forces being then in the same plane, it is 

 necessary, in order that they may be in equilibrium, that some one 

 of them, the force p for example, should produce two efforts, the 

 one equal and opposite to the force q, and the other equal and op- 

 posite to the force r. Now if, after having produced rA, q A, we 

 take any line AD to represent the force p, and upon AD as a diag- 

 onal we construct the parallelogram ACDB, the two sides AB, AC, 

 will represent two forces, which acting conjointly according to 

 these directions, would produce the same effect as the force p. 40. 

 Accordingly, AB, AC, are the efforts that p actually opposes to the 

 Mech. 11 



