A Treatise on the Theory of Screws

400] THE THEORY OF SCREWS IN NON-EUCLIDIAN SPACE. 439 whence we deduce Xcf (X) = (/a). As X. and p are perfectly independent, this equation can only subsist by assuming for </> a form such that (X) = 77, where H is independent of X. Whence we obtain A- and, by integration, < /> (X) = H log X + constant. The intervene is now readily determined, for < /> (X) - </> (u.) = H log X - H log p = H log -. We therefore obtain the following important theorem which is the well- known* basis of the mensuration of non-Euclidian space:— Let x2, xs, xit and ylt y2, y3, yt be the two objects at infinity on a range, and let + x2 + Xy2, x3+Xy3, + Xyit and x. + py^, x2 +py2, x3 +py3, Xt + pyt be any two other objects on the range, then their intervene will be expressed by H (log X - log p), where H is a constant depending upon the adopted units of measurement. It will be useful to obtain the expression for the intervene in a rather more general manner by taking the equation in Å, and p, for objects at the intervene 8, as ylX/z, + Bx + Cp + D = 0. Let X' and X" be the two roots of this equation when p is made equal to X. It follows that what we have just written may be expressed thus: — Xp + X (0 - - jX") + P (“ 0 - i1' ~ + X'X" = °- For, if X = p, this is satisfied by either X' or X", while 0 disappears. 0 is, of course, a function of the intervene, and it is only through 0 that the inter- vene comes into the equation. By solving for 0, we find X/t-i(X'+X")(X + /*) + VX" 0 ~ ff-X ’ * Professor George Bruce Halsted remarks in Science, N. S., Vol. x„ No. 251, pages 545—557, October 20 1899, that “Roberto Bonola has just given in the Bolletino di Bibliografi« e Stona della Science Matematiche (1899) an exceedingly rich and valuable Bibliografia sui Fondamenti, della Geometria in relations alia Geometria non-Euolidea in which he gives 353 tatles.