Rope Machine. 89 



or, 



q : p : : sin r AC ; sin q A r, 

 which agrees also with the proposition above referred to. 143. 



154. We shall now inquire into the changes that take place 

 in the communication of the action exerted by the powers in 

 consequence of the gravity of the cords. 



Let there be any number of powers applied to the same cord 

 gABC-sr drawn at its two extremities by the two powers p, w,jpi gt ($3. 

 and retained at two fixed points p and *. 



If we produce the two extreme cords Q A^ C, until they 

 meet in F, it is evident that the resultant of the particular ten- 

 sions of these two cords must pass through the point F. And 

 since an equilibrium is supposed, the resultant of the three pow- 

 ers j, q, r, and of the tensions of the two intermediate cords AB, 

 BC, must also pass through the point F; since, in order to an 

 equilibrium, this resultant must be equal and directly opposite to 

 the resultant of the tensions of the two cords QA,vrC. But the 

 resultant of the three powers, and of the tensions of the two in- 

 termediate cords, is nothing but the resultant of the three powers 

 simply, because each of the two cords AB, C, has by itself no 

 action whatever, and consequently no effect upon any part of 

 the system. Therefore the resultant of all the powers jo, <?, r, 

 applied to the cord, passes through the point of meeting V of the 

 two extreme cords. 



It has been shown how this resultant may be determined ; 42, &c, 

 but if the cords are parallel, as is the case when the powers 

 p, q, r, are weights, since their resultant cannot but be parallel 

 to them, its direction is found very simply by drawing through 

 the point V a line parallal to one of the directions of these 

 weights, that is, by drawing a vertical or perpendicular line. 



Accordingly, let there be any number of weights applied to 

 the same cord Q ABCD v destitute of gravity. The two extreme Fig. 64. 

 cords being produced, and a vertical VX being drawn through 

 their point of meeting, we can reduce the equilibrium of the 

 whole system to the case in which the three powers, applied to 

 three cords, are united by the knot F, and in which the power 

 directed according to XVZ is the sum of the weights. We 

 Meek. 12 



