A Treatise on the Theory of Screws

348] THE THEORY OF SCREW-CHAINS. 381 with similar values for Sy, Sy', Sø, X/ then the above equations give 2a/ = a 'tæ + b Xy + c 'Zz, S,y' = a S« + b’ Xy + c Xz, 1z = a"Xv + b"ty + c"1z. If the system of forces in the first plane equilibrate, the following con- ditions must be satisfied: S® = 0, 1y = 0, = 0, and from the equations just written, these, involve XX =0, Xy'=0, 0, whence the corresponding system in the other plane must also equilibrate. To determine the correspondence it will be necessary to know only the three forces in the second plane which correspond to three given forces in the first plane. We shall then have the nine equations which will be sufficient to determine the nine quantities a, b, c, &c. It appears, from the form of the equations, that the ratio of the intensity of a force to the intensity of the corresponding force is independent of those intensities, i.e. it depends solely upon the situation of the lines in which the forces act. Take any four straight lines in one system, and let four forces, Xlt Xit Xs, Xlt on these four straight lines equilibrate. It is then well known that each of these forces must be proportional to certain functions of the positions of these straight lines. We express these functions by tIj, j42, A3, At. The four corresponding forces will be Xi, X2\ X-i, Xi, and as they must equilibrate, they must also be in the ratio of certain functions Ai, Ai, Ai, Ai of the positions. We thus have the equations Z1=^2=X3=X Ai A. A3 Ai ’ -41 A2 A3 Ai We can select the ratio of to Xi arbitrarily : for example, let this ratio be p\ then Xi Ai whence the ratio of X,2 to Xi is known. Similarly the ratio of the other intensities X3 : Xi, and X4 : Xi is known. And generally the ratio of every pair of corresponding forces will be determined.