A Treatise on the Theory of Screws

CHAPTER XXVI . AN INTRODUCTION TO THE THEORY OF SCREWS IN NON-EUCLIDIAN SPACE. 396. Introduction. The Theory of Screws in non-Euclidian space is a natural growth from some remarkable researches of Clifford* in further development of the Theory of Riemann, Cayley, Klein and Lindemann. I here give the in- vestigation sufficiently far to demonstrate two fundamental principles (§§ 427, 434) which conduct the theory to a definite stage at which it seems convenient to bring this volume to a conclusion. I have thought it better to develop from the beginning the non-Euclidian geometry so far as we shall at present require it. It is thus hoped to make it intelligible to readers who have had no previous acquaintance with this subjectf. I give it as I have worked it out for my own instruction+. It is indeed characteristic of this fascinating theory that it may be surveyed from many different points of view. 397. Preliminary notions. Let «j, x.,_, xs, be four numerical magnitudes of any description. We may regard these as the co-ordinates of an object. Let y1; y2, y3, yt be the co-ordinates of another object, then we premise that the two objects will be identical if, and only if ?a ~y* y*~ y* All possible objects may be regarded as constituting a content. * “Preliminary Sketch of Biquatermons,” Proceedings of the London Mathematical Society, Vol. iv. 381—395 (1873). See also “On the Theory of Screws in a Space of Constant Positive Curvature,” Mathematical Papers, p. 402 (1876). Clifford’s Theory was much extended by the labours of Buchheim and others ; see the Bibliographical notes. + We are fortunately now able to refer English readers to a Treatise in which the Theory of non-Euelidian space and allied subjects is presented in a comprehensive manner. Whitehead, Universal Algebra, Cambridge, 1898. $ Trans. Roy. Irish Acad., Vol. xxvm. p. 159 (1881), and Vol. xxix. p. 123 (1887). c. 28