Rope Machine. S3 



from which we should readily determine the side DB, or the force 

 r, and the angle EDA, or its equal r AS, formed by the direc- 

 tions of r and p. If the angles formed by the directions of the Trig. 36. 

 three forces were given 1 , we could not thence determine the ab- 

 solute values of the three forces, but only their ratio to each 

 other. In all other cases, the proposition above established will 143- 

 be sufficient for a complete solution, when three things only are 

 given. 



145. If instead of having two forces, q and r, attached to two 

 cords, these two cords were firmly fixed at q and r, or at any 

 points respectively in their directions, AB, AC, would express 

 the efforts supported by these points. 



146. We have supposed the three cords firmly attached by Fig. '55. 

 a knot A. But if the cord to which the power p is applied had F<i g- 57 ' 

 a ring at its extremity A, through which the cord q A r passed, 



we should not be able to assign the directions of the three cords. 

 Indeed it is not sufficient, in this case, that the effort AB has the 

 direction qA, and is equal to the force </, and that AC has 

 the direction rA, and is equal to r ; it is necessary, further, that 

 the ring should not slip upon the cord q A r, which requires that 

 the angle q AS should be equal to SA r ; that is, that the power 

 p should be directed in such a manner, as to bisect the angle 

 q A r. But we have always 



p : q : r :: sin qAr: sin rAS : sin q AS ; 

 and as r AS = q AS = J q A r, this series of ratios becomes 



p : q : r : : sin q A r : sin \ q A r : sin | q A r ; 

 so that the two powers -q and r are equaL 



147. The same result would follow, if the cord q A r, drawn 

 by the two powers r, 7, passed over a fixed point A. The two 

 powers r, q, must be equal, and the force exerted by them upon 

 the fixed point will be directed in such a manner as to bisect the 

 angle q A r, and its magnitude will be with respect to each of 

 these two powers, as the sine of q A r is to the sine of half q A r. 



148. The foregoing articles being well understood, it will be 

 easy to determine the conditions of equilibrium among as many 

 powers as we choose to employ, applied to different cords, and 

 united by the same or by different knots. 



