A Treatise on the Theory of Screws

515 BIBLIOGRAPHICAL NOTES. Form now the determinant A— If A„ = 0, tl 2, 1 3, 1 4, 1 5, 1 6, 1 le lines 1, 2 1, 3 2, 3 3, 2 4, 2 4, 3 5, 2 5, 3 6, 2 6, 3 are in involution. 1, 4 1, 5 1, 6 2, 4 2, 5 2, 6 3, 4 3, 5 3, 6 4, 5 4, 6 5, 4 5, 6 6, 4 6, 5 Considering only the figi ires 1, 2, 3, 4, 5 the determinant As can be formed. If A,; = 0 and A5 = 0, the five lines 1, 2, 3, 4, 5 are in involution. If all the other minors are zero, the six lines will intersect a single transversal. If A5 = 0, without any other condition, the five lines 1, 2, 3, 4, 5 intersect a single transversal. If A4 = 0, without any other condition, the lines 1, 2, 3, 4 have but one common transversal (Cayley). A determinant can be found which is equal to the square root of A6. Grassmann (H.)—Die Ausdehnungslehre. Berlin (1862). This remarkable work, a development of an earlier volume (1844), by the same author, contains much that is of instruction and interest in connection with the present theory. A system of n, numerically equal, “ Grössen erster Stufe,” of which each pair are “normal,” is discussed on p. 113. A set of co-reciprocal screws is a particular case of this very general conception. The “ inneres Produkt ” of two “ Grössen ” divided by the product of their numerical values, is the cosine of the angle between the two “ Grössen.” If a, b, c, ... be normal, and if k, I be any two other “Grössen,” then cos tkl = cos zak cos zal + cos zbk . cos zbl, + &c. (p. 139). Here we have a very general theory, which includes screw co-ordinates as a particular case. In a note on p. 222 the author states that the displacement of a body in space, or a general system of forces, form an “allgemeine räumliche Grösse zweiter Stufe.” The “kombinatorisches Produkt” (p. 41) of n screws will contain as a factor that single function whose evanescence would express that the n screws belonged to a screw system of the (n- l)th order. Plücker (J.)—On a new geometry of space. Phil. Trans., Vol. civ., pp. 725—791. 1865. In this paper the linear complex is defined (p. 733). Some applications to optics are made (p. 760); the six co-ordinates of a line are considered (p. 774); and the applications to the geometry of forces (p. 786). This is of importance for our purpose because the linear complex may be also defined with perfect generality as the axes of all the screws of any stated pitch which belong to a 5-system. The relation of the linear geometry to Dynamics is developed in the Theory of Screws. Hamilton (Sir W. R.)—Elements of Quaternions. Dublin, 1866. In Art. 416 the equation 2 V (a — y) ß - 0, is regarded as the single equation of equilibrium when it is satisfied for all values of y, the vector to an arbitrary point C in space. In general if y is not supposed to vary in this arbitrary manner, the equation is that of the central axis. 33—2