A Treatise on the Theory of Screws

401] THE THEORY OF SCREWS IN NON-EUCLIDIAN SPACE. 441 first time. No doubt we find the intervene to be the logarithm of an an- harmonic ratio of four quantities, but these quantities are not distances nor are they quantities homologous with the intervene. They are simply numerical. The four numbers, X, /z, X', X" are merely introduced to define four objects, one of them being, «i + Xy, x.,+ Xy2, x3 + Xys, xt + Xyt, and the others are obtained by replacing X by /z, V, X”, respectively. All we assert is, that if we choose to call the two objects defined by X' and X" the objects at infinity, and that if we desire the intervene between the objects X and /z to possess the properties that we have already specified, then the only function possible will be the logarithm of the anharmonic ratio of these four numbers. The word anharmonic is ordinarily applied in describing a certain function of four collinear points. In the more general sense, in which we are at this moment using the word, it does not relate to any geometrical or spacial relation whatever; it is a purely arithmetical function of four abstract numbers. We may also observe that the relation between 0 and the intervene 3 is given by the equation a , 1 0 = |(V-V)V- eH-l and the expression of the intervene as a function of 0; that is, the expression F(0) is 8 = Hk>g 0 + ^(X,z — X') 401. Another process. We may also proceed in the following manner. Let us denote the values of X for the infinite objects on the range by peie and pe~ie. If then X, p be two parameters for two objects at an intervene 8, we must have (p. 439) Xp + X (e - p cos 0) + p (- e - p cos 0) + p2 = 0. Solving for e, we have Xp — p cos 0(X+ p) + p- e _ ——————————- . /X — A. The intervene 8 must be some function of e, whence