A Treatise on the Theory of Screws

348] THE THEOKY OF SCREW-CHAINS. 379 a case of “Rational Transformation” (see Salmon’s Higher Plane Curves, Chap. VIII.). The theory is however here much simplified. In this case none of the special solutions are admissible which produce the critical cases. Consider the equations U2=0, P5 = 0. They will give a number of systems of values for 0lr 0B equal to the product of the degrees of U2, UB. Each of these 0 screws would be a correspondent to the same </> screw 1, 0, 0, 0, 0. But in the problems before us this <£ as every other </> can have only one correspondent. Hence all the functions U\, U2, etc. tf/, U2', etc. must be linear. We may express the first set of equations thus: = (11) 0, + (12) 0, + (13) 03 + (14) 0t + (15) 05, < />3 = (21) 0, + (22) 03 + (23) 03 + (24) 0t + (25) 0B, < />3 = (31) 0. + (32) 02 + (33) 03 + (34) 04 + (35) 0B, & = (41) 0, + (42) 02 + (43) 03 + (44) 0t + (45) 0B, = (51) 0, + (52) 02 + (53) 03 + (54) 0, + (55) 0B. For the screw </> to be known whenever 0 is given, it will be necessary to determine the various coefficients (11), (12), &c. These are to be determined from a sufficient number of given pairs of corresponding screws. Of these co- efficients there are in all twenty-five. If we substitute the co-ordinates of one given screw 0, we have five linear equations between the co-ordinates. Of these equations, however, we can only take the ratios, for each of the co- ordinates may be affected by an arbitrary factor. Each of the given pairs of screws will thus provide four equations to aid in determining the co- efficients. Six pairs of screws being given, we have twenty-four equations between the twenty-five coefficients. These will be sufficient to determine the ratios of the coefficients. We thus see that by six pairs of screws the homography of two five-systems is to be completely defined. To any seventh screw on one system corresponds a seventh screw on the other system, which can be constructed accordingly. 348. Application of Parallel Projections. It will, however, be desirable at this point to introduce a somewhat different procedure. We can present the subject of homography from another point of view, which is specially appropriate for the present theory. The notions now to be discussed might have been introduced at the outset. It was, however, thought advantageous to concentrate all the light that could be obtained on the subject; we therefore used the point-homography of lines, of planes, and of spaces, so long as they were applicable. The method which we shall now adopt is founded on an extension of what are known as “parallel projections” in Statics. We may here recall