A Treatise on the Theory of Screws

516 THE THEORY OF SCREWS. (0, 0), (0, 1) ... (1, 0), (1, 1) ... * We may remark that since the moment of two lines is the virtual coefficient of two screws of zero pitch, these equations are given at once by virtual velocities, if we rotate the body round each of the forces in succession. Spottiswoode (W.)—Note sur Véquilibre des forces dans Vespace. Comptes Rendus; Vol. Ixvi., pp. 97-103 (January, 1868). If P^... Pn_^ be n forces in equilibrium, and if (0, 1) denote the moment of Po, P1, then the author proves* that A(0, l) + A.(0, 2) + ...=0, p, (1,0)+ + P2 (1, 2) + ... = 0, Po(2, 0) + P1(2, 1)+ +...=0. As we have thus n equations to determine only the relative values of n quantities, the redundancy is taken advantage of to prove that p 2 p 2 [0, 0] “ [1, where [0, 0], [1, 1], &c., are the coefficients of (0, 0), (1, 1), &c., in the determinant He then considers the quaternion q = — @, already mentioned (p. 512), and introduces the new quaternion Q = = c + y. The scalar c (the pitch) is inde- pendent of the assumed origin, and the vector y is the vector to a definite point C on the central axis. This point does not vary with the position of the assumed origin, and is called the “Centre of the System of Forces.” When the forces are all parallel G coincides with the centre of the parallel forces. In general Ttaß = V(c2 + 7’/) Ttß, or the tensor of the total moment is constant for all points 0 situated on a sphere whose centre is C, and becomes a minimum when 0 coincides with C. In Art. 396 Hamilton says “ the passage of a right line from any one given position in space to any other maybe conceived to be accomplished by a sort of screw motion," and on these kinematical lines he worked out his theory of the “Surface of Emanants,” generated by a line moving according to some given law and constantly intersecting a given curve in space. Plücker (J.)—Fundamental views regarding mechanics. Phil. Trans. (1866); Vol. clvi., pp. 361—380. The object of this paper is to “ connect, in mechanics, translatory and rotatory movements with each other by a principle in geometry analogous to that of re- ciprocity.” One of the principal theorems is thus enunciated:—“Any number of rotatory forces acting simultaneously, the co-ordinates of the resulting rotatory force, if there is such a force, if there is not, the co-ordinates of the resulting rotatory dyname, are obtained by adding the co-ordinates of the given rotatory forces. In the case of equilibrium the six sums obtained are equal to zero.”