A Treatise on the Theory of Screws

518 THE THEORY OF SCREWS. have a common surface of singularities where A is a variable parameter. If the roots A1( &c. be known, we have a set of'quasi elliptic co-ordinates for the line x. (Compare § 234.) It is in this memoir that we find the enunciation of the remarkable geometrical principle which, when transformed into the language and conceptions of the Theory of Screws, asserts the existence of one screw reciprocal to five given screws. (§ 25.) Klein (F.)—Die allgemeine lineare Trangformation der Linien Coordinaten. Math. Ann.; Vol. ii., pp. 366-371 (August 4, 1869). Let Z7,, ... If. denote six linear complexes. The moments of a straight line, with its conjugate polars with respect to If, ... If, are, when multiplied by certain constants, the homogeneous co-ordinates of the straight line, and are denoted by ajj, ... xe. Arbitrary values of aij, &c., do not denote a straight line, unless a homogeneous function of the second degree vanishes*. If this condition be not satisfied, then a linear complex is defined by the co-ordinates, and the function is called the invariant of the linear complex. The simultaneous invariant of two linear complexes is a function of the co-ordinates, and is equal to A sin <f> — (K + K') cos </>, where K and K' are the parameters of the linear complexes, A the perpendicular distance, and </> the angle between their principal axes. The co-ordinates of a linear complex are the simultaneous invariants of the linear complex with each of six given linear complexes multiplied by certain constants. The six linear complexes can be chosen so that each one is in involution with the remaining five. The reader will easily perceive the equivalent theorems in the Theory of Screws. K and A' ai’e the pitches, and the simultaneous invariant is merely double the virtual coefficient with its sign changed. Zeuthen (H, G.)—Notes sur un Systeme de coordonnées linéaires dans Vespace. Math. Ann.; Vol. i., pp. 432-454 (1869). The co-ordinates of a line are the components of a unit force on the line decom- posed along the six edges of a tetrahedron. These co-ordinates must satisfy one condition, which expresses that six forces along the edges of a tetrahedron have a single resultant force. The author makes applications to the theory of the linear complex. Regarding the six edges as screws of zero pitch, they are not co-reciprocal. It may, however, be of interest to show how these co-ordinates may be used for a purpose different from that for which the author now quoted has used them. Let the virtual coefficients of the opposite pairs of edges be L, M, N. If the co- ordinates of a screw with respect to this system be 0, ... 06, then the pitch is (L0& + M0S04 + N0506), and the virtual coefficient of the two screws </>, 0 is (0^, + 0^) + J Jf + 0^3) + %N (05</>6 + 06</>6). Battaglini (G.)—Suite serie die sistemi di forze. Napoli Rendiconto, viii., 1869, pp. 87-94. Giornale di Matemat, x., 1872, pp. 133-140. This memoir deserves special notice in the history of the subject inasmuch as alrBädy remarked in § 13 it contains the earliest announcement of the dynamical significance of the cylindroid. Battaglini here shows that the cylindroid is the locus of the screws on which lie the wrenches produced by the composition of two variable forces on two fixed directions. See also p. 520. This equation expresses that the pitch of the screw denoted by the co-ordinates is zero.