A Treatise on the Theory of Screws

77] THE EQUILIBRIUM OF A RIGID BODY. 67 77. Remark on systems of Linear Equations. Let a right line be, as usual, represented by the two equations Ax + By + Cz + D = 0, A'x + B'y+ C'z+ D'= 0. There are here six independent constants involved, while a right line is completely defined by four constants. The fact of course is that these two equations not only determine the right, line on which our attention is fixed, but they also determine two planes through that line. Four constants are needed for the straight line and one more for each of the planes, so that there are six constants in all. If we are concerned with the straight line only the intrusion of two superfluous constants is often inconvenient. We can remove them by first eliminating y and then x, thus giving the two equations the form x = Pz + Q, y = P'z + Q'. We have here no more than the four constants P, Q, P', Q', which are indis- pensable for the specification of the straight line. Of course it may be urged that these equations also represent two planes. No doubt they do, but the equation z = Px + Q is a plane parallel to the axis of y, which is absolutely determined when the straight line is known. The plane Ax + By + Cz + D = 0 may represent any one of the pencil of planes which can be drawn through the straight line. Analogous considerations arise when the screws of an n-system are represented by a series of linear equations. We commence with the case of the two-system, in which of course the screws are limited to the generators of a cylindroid. Let 32, 32,... 3C, be the co-ordinates of a screw 3 referred to any six screws of reference. Let these co-ordinates satisfy the four linear equations Aj 3, + A232 + ... + = 0, Bi 0, + B2 32 + ... + B,s 3,t = 0, (j\3t + (j232 + ... + Ce3e — 0, Di 3i + D., 32 + ... + De 3e = 0, where Alt A2, ..., Bt, B2, ... C2 and Dlt D.2, ... are constants. Then it is a fundamental part of the present Theory that the locus so defined is a cylindroid (§ 76). 5—2