A Treatise on the Theory of Screws

■■■■■■■■■■■■■■■■■OHBSkESSSESS 238 THE THEORY OF SCREWS. [225, If B = 0 the two roots of the equation in pe will be equal, but with opposite signs; as cos X, cos/z, cos v enter to the second degree in B it follows that through any point in space as vertex a cone of the second degree can be drawn such that each generator of this cone when the proper pitch is assigned to it will equally belong to the quadratic five-system, whether that pitch be positive or negative. If -B = 0 and C = 0, then both values of p6 must be zero. Regarding x y y > z as fixed, each of these equations will correspond to a cone with vertex at x, y, z ; these cones will have four common generators, and hence we see that through any point in space four straight lines can in general be drawn such that with the pitch zero but not with any other pitch, these screws will be members of a given quadratic five-system. 226. Polar Screws. The general discussion of the quadratic screw-systems is a subject of interest both geometrical and physical. We shall here be content with a few propositions which are of fundamental importance. Let as before ^ = 0 be the homogeneous relation between the co-ordinates 01, ...,0„+1 of the screws which constitute a quadratic n-system. Let y and £ denote any two screws other than d and chosen from the enclosing n-system, from which the screws of the quadratic n-system are selected by the aid of the condition Ue = 0. If then we adopt the fertile method of investigation introduced by Joachimsthal, we shall substitute in mUt = 0 for ,@n+i the respective values Uh + ly.2 + mf,......lyn+1 + mt;n+1. The result will be Z2 + Im + m?U( = 0, where Uv(=^+... + . Solving this quadratic equation for I m we obtain two values of this ratio and hence (§ 119) we deduce the following theorem. -^■ny cylindroid of a given (n + system will possess generally two screws belonging to every quadratic n-system which the given (n + l)-system encloses. If the two screws y and f had been so selected that they satisfied the condition u,=i:~i+ + h -o