.* Rope Machine. 87 



to more than four. But if the number does not exceed four, the 

 directions being given, the ratios that must exist among the forces 

 applied to these cords respectively, are determinate. For through 

 any two of these cords, as Ap, A w, a plane may be supposed to Fig. 6U 

 pass, which produced would meet the plane r A q of the two 

 other cords in some line DAE, the position of which is determin- 

 ed by the directions of the four powers. Then, the direction &A 

 being produced, and JIB being taken to represent the power p, 

 if upon AB as a diagonal, and upon the directions AD, AC, as 

 sides, we construct the parallelogram DACB, AC will represent 

 the value of the power , and AD the effort made by the power 

 p against the two powers q and r acting conjointly. Accordingly, 

 having produced qA and rA (which are ift the same plane with 

 AD) to F and G, if upon AD as a diagonal, and upon AF, AG, 

 as sides, we construct the parallelogram AFDG, AF, AG, wiH 

 represent the values belonging to the two powers q and r. 



1 53. Finally, whatever the case may be, whether the cords 

 are in the same plane or not, as a state of equilibrium requires 

 that each knot should remain immoveable, if the force or tension of 

 each cord, applied to the same knot, be decomposed into three 

 other forces parallel to three rectangular co-ordinates, it is nec- 

 essary with respect to each- knot that the sum of the forces paral- 

 lel to each of these lines should be equal to zero; (it being well un- 

 derstood that by the word sum, as here used, is meant the sum of 129 - 

 the forces that act in one direction, minus the sum of those which 

 act in the opposite direction). If the cords united by the same 

 knot, were in the same plane, it would be sufficient to decompose 

 the tensions respectively into two forces parallel to two lines 

 perpendicular to each other, and drawn in the same plane. This 

 method would give in every case all the conditions of equilibri- 

 um, the cords being supposed to be firmly connected among 

 themselves. 



To give a simple example of this method, let it be proposed 

 to find the ratio of three powers in equilibrium by means of three 

 cords united by the same knot. 



Let us suppose for a moment that these three powers admit F 'g- 6a > 

 of being represented by the three lines AG, AB, AF, and in or- 

 der to abridge the decomposition, let the two powers p, q* be 



