A Treatise on the Theory of Screws

264 THE THEORY OF SCREWS. [246- If these six equations be solved for a1; ... aß we must have cti = . ß6) ae = Fe(filf ... fif As a single a is to correspond to a single fi, and vice versa, these equations must be linear: whence we have the following important result:— In two homographic screw systems the co-ordinates of a screw in one system are linear functions with constant coefficients of the co-ordinates of the corre- sponding screw in the other system. If we denote the constant coefficients by the notation (11), (22), &c., then we have the following system of equations:— & = (11) öj + (12) a.2 + (13) a3 + (14) a4 + (15) a5 + (16) a6, fi2 = (21) a, + (22) a2 + (23) a3 + (24) a4 + (25) a5 + (26) a6, fit = (61) a, + (62) a2 + (63) a3 + (64) a4 + (65) a5 + (66) a6. 247. The Double Screws. It is now easy to show that there are in general six screws which coincide with their corresponding screws; for if ß1 — palt ß2 = pa.,, &c., we obtain an equation of the sixth degree for the determination of p. We therefore have the following result:— In two homographic screw systems six screws can in general be found, each of which regarded as a screw in either system coincides with its correspondent in the other system. 248. The Seven Pairs. In two homographic rows of points we have the anharmonic ratio of any four points equal to that of their correspondents. In the case of two homographic screw systems we have a set of eight screws in one of the systems specially related to the corresponding eight screws in the other system. We first remark that, given seven pairs of corresponding screws in the two systems, then the screw corresponding to any other given screw is deter- mined. For from the six equations just written by substitution of known values of alt ... a6 and filt ... fi6, we can deduce six equations between (11), (12), &c. As, however, the co-ordinates are homogeneous and their ratios are alone involved, we can use only the ratios of the equations so that each pair of screws gives five relations between the 36 quantities (11), (12), &c. The