A Treatise on the Theory of Screws

514 THE THEORY OF SCREWS. Cayley (A.)—On a new analytical representation of curves in space. Quarterly Mathematical Journal; Vol. iii., pp. 225-236 (1860). Vol. v., pp, 81-86 (1862). Coll. Math. Papers, Vol. iv. pp. 446-455, 490-494. In this paper the conception of the six co-ordinates of a line is introduced for the first time. This is of importance in connection with our present subject because the six coordinates of a screw may be regarded as the generalization of the six coordinates of a straight line. If a,,... a8 be the six coordinates of a screw then when we express that its pitch is zero by the condition pya^+ ... + fta62=0 we obtain the coordinates of a straight line. This is perhaps the most symmetrical form of the quadratic condition which must subsist between six quantities consti- tuting the coordinates of a line. In Cayley’s system it is given by equating the sum of three products to zero. Sylvester (J. J.)—Sur I’involution des lignes droites dans Vespace, considérées comme des axes de rotation. Paris, Comptes Rendus: vol. Iii., pp. 741—746 (April, 1861). Any small displacement of a rigid body can generally be represented by rota- tions about six axes (Möbius). But this is not the case if forces can be found which equilibrate when acting along the six axes on a rigid body. The six axes in this case are in involution. The paper discusses the geometrical features of such a system, and shows, when five axes are given, how the locus of the sixth is to be found. Möbius had shown that through any point a plane of lines can be drawn in involution with five given lines. The present paper shows how the plane can be constructed. All the transversals intersecting a pair of conjugate axes are in involution with five given lines. Any two pairs of conjugate axes lie on the same hyperboloid. Two forces can be found on any pair of conjugate axes, which are statically equivalent to two given forces on any other given pail’ of conjugate axes. In presenting this paper M. Chasles remarks that Mr Sylvester’s results lead to the following construction :—Conceive that a rigid body receives any small displace- ment, then lines drawn through any six points of the body perpendicular to their trajectories are in involution. M. Chasles also takes occasion to mention some other properties of the conjugate axes. Sylvester (J. J.)—Note sur I’involution de six lignes dans I’espace. Paris, Comptes Rendus; vol. Iii., pp. 815-817 (April, 1861). The six lines are 1, 2, 3, 4, 5, 6. Let the line i be represented by the equations atx + bty + ctz + dtu = 0, aix + ßiy + yt* + = 0, and let i, j represent the determinant «4 bt ct dt ai ßi yi aJ Cj aj ßj yj fy-