86 Statics. 



151. When the number of cords united by the same knot 

 exceeds three, being in the same plane, or when,'being in differ- 

 ent planes, the number exceeds four, the directions being given, 

 the ratios of the powers and of the tensions of the cords are not 

 absolutely determinable ; that is, if a certain number of powers 

 (not less than those just stated) be in equilibrium according to 

 known directions, we can substitute instead of them a like number 

 of other powers directed in the same manner, but which, having 

 very different ratios among themselves, are notwithstanding in 



Fig. 60. equilibrium. If, for example, the four cords A p, A q, A r, A w, 

 are all in the same plane, having taken AB to represent the 

 force p, and having produced the cord ?/ A to C, we suppose the 

 effort AB composed of two others AC, AD, the first of which is 

 qual and directly opposite to the power vr, nothing can be infer- 

 red from the direction AD of the action that is to oppose itself to 

 the effort of the two powers q, r ; nothing, I say, can be inferred 

 from this direction, except that, produced, it must pass into the 

 angle q A r ; a condition which may evidently be satisfied in an 

 infinite number of ways. Accordingly, if AD be drawn in any 

 manner whatever, within the angle formed *by A q and A r pro- 

 duced, and we construct upon AB as a diagonal, and upon the 

 directions AC, AD, as sides, the parallelogram ACBD, and then 

 upon AD as a diagonal, and upon q A, r A, produced, as sides^ 

 we construct also the parallelogram AEDF, AB being taken to 

 represent the value of p, AC may be taken to represent that of 

 -, AF that of r, and AE that of q. This is evident, because the 

 force AB is equivalent to the two forces AC, AD, the first of 

 which, in order to be in equilibrium with &, must be equal to w, 

 and the second AD is equivalent to the two forces AF, AE, which 

 to be in equilibrium with r and q, must be equal to r and q res- 

 pectively. But it will be seen at the same time, that by giving 

 io AD a different direction, AC, AF, AE, will have different val- 

 ues, but such notwithstanding that being taken to represent the 

 powers acting in these directions, an equilibrium would be pro- 

 duced ; so that in this case, the directions remaining the same, 

 there is an infinite variety of ways in which an equilibrium can 

 be effected among the powers in question. 



152. The problem is of a similar character \vhen the cords 

 proceeding from the same knot, are in different planes, and amount 



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