A Treatise on the Theory of Screws

266 THE THEORY OF SCREWS. [249- When these substitutions are made in the determinants it is plain that they still vanish; we hence have the important result that The screws corresponding homographically to the screws of an n-system form another n-system. Thus to the screws on a cylindroid will correspond the screws on a cylindroid. It is, however, important to notice that two reciprocal screws have not in general two reciprocal screws for their correspondents. We thus see that while two reciprocal screw systems of the rath and (6 — n)th orders respectively have as correspondents systems of the same orders, yet that their connexion as reciprocals is divorced by the homographic transforma- tion. Reciprocity is not, therefore, an invariantive attribute of screws or screw systems. There are, however, certain functions of eight screws analogous to anharmonic ratios which are invariants. These functions are of considerable interest, and they are not without physical significance. 250. Analogy to Anharmonic Ratio. We have already (§ 230) discussed the important function of six screws which is called the Sexiant. This function is most concisely written as the determinant (a1(S2y334e6^6) where a, ß, y, 8, e, are the screws. In Sylvester’s language we may speak of the six screws as being in involution when their sexiant vanishes. Under these circumstances six wrenches on the six screws can equilibrate; the six screws all belong to a 5-system, and they possess one common reciprocal. In the case of eight screws we may use a very concise notation; thus 12 will denote the sexiant of the six screws obtained by leaving out screws 1 and 2. It will now be easy to show that functions of the following form are invariants, i.e. the same in both systems:— 12. 34 13.24 It is in the first place obvious that as the co-ordinates of each screw enter to the same degree in the numerator and the denominator, no embarrassment can arise from the arbitrary common factor with which the six co-ordinates of each screw may be affected. In the second place it is plain that if we replace each of the co-ordinates by those of the corresponding screw, the function will still remain unaltered, as all the factors (11), (22), &c., will divide out. We thus see that the function just written will be absolutely unaltered when each screw is changed into its corresponding screw. By the aid of these invariant functions it is easy, when seven pairs of screws are given, to construct the screw corresponding to any given eighth