A Treatise on the Theory of Screws

528 THE THEORY OF SCREWS. Buchheim (A.)—On the Theory of Screws in Elliptic Space. Proceedings of the London Math. Soc., Vol. xiv., p. 83; Vol. xvi., p. 15; Vol. xvii., p. 240; Vol. xviii., p. 88. In these papers the methods of the Ausdehnungslehre of Grassmann have been applied to the Biquaternions of Clifford. Reference should also be made to another paper by the same author, “A Memoir on Biquaternions.’’ (American Journal of Mathematics, Vol. vii., No. 4, p. 23, 1884.) If A, B be two biquaternions they determine a linear singly infinite series of biquaternions XA + yB when Å, p are scalars : this set is called a cylindroid, so that if C is any biquaternion of the cylindroid A, B we have C = Å.A + pß. A remarkable investigation of the equation to this surface in elliptic space is given, and a generalization of the plane representation of the cylindroid is shown. In these writings it is the methods employed that are chiefly noticeable. We find however much more than is implied by the modest disclaimer of the lamented writer, who in the last letter I had from him says, “I have been but slaying the slain, i. e. discovering over again results obtained by you and Clifford.” Segre (0.)—Sur une expression nouvelle du moment mutuel de deux complexes linéaires. Kronecker’s Journal, pp. 169-172 (1885). A remarkable form for the expression of the virtual coefficient of two screws on a cylindroid is given in this paper. Translated into the terminology of the present volume we can investigate Segre’s theorem as follows. Let two screws on the cylindroid make angles 0, <f>, with one of the principal screws, while the zero pitch screws make angles + a, — a. Let p be the anharmonic ratio of the pencil parallel to these foui’ screws so that sin (Ö — a) sin (</> + a) sin (0 + a) ’ sin (</> — a) ’ Then as usual Pe - Po. + m cos 2Ö, o =p0 + m cos 2a; whence pe = 2 m sin (a — 6) sin (a + 0), p,(, = 2m sin (a — </>) sin (a + f); whence 4m2 sin2 (0 — a) sin2 (</> + a) - ppep<p Thus 2 m sin (0 - a) sin (</> + a) = Jp Jpep^, 2 m sin (0 + a) sin (</> - a) = Jp~' Jp^ ; adding, we easily obtain = i (Jp + Vp-1) JPeP*- If therefore we make P — ®21'> we have as the result Segre’s theorem that ^Jps'JN cos e- D’Emilio (R).—Gli assoidi nella statica e nella cinematica. Nota su la teoria delle dinami. Atti del Reale Istituto Veneto di scienze, (6) iii. 1135-1154 (1885). This is an account of the fundamental laws of the different screw-systems.