A Treatise on the Theory of Screws

225J FREEDOM OF THE FOURTH ORDER. 235 degree seems not to be justified. It is however shown in § 202 that this hyperboloid is really not more than a part of the locus. There are also four imaginary planes which with the hyperboloid complete the locus, and the combination thus rises to the sixth degree. 225. The Quadratic Systems of Higher Orders. If we had taken n = 3, then of course the quadratic three-system would mean the collection of screws whose four co-ordinates satisfied an equation which in form resembles that of a quadric surface in quadriplanar co- ordinates. A definite number of screws belonging to the quadratic three- system can in general be drawn through every point in space. We shall first prove that the number of those screws is six. Let 04, ...,06 be the co-ordinates of any screw 0 referred to a canonical co-reciprocal system. Then if a!, y, z' be a point on 0, we have (§ 43) (0, + Øs) y' - (03 + h) = a (0X - 02) - pe (01 + 0-f (01 + 0-1) - (#s + Øs) % = b (03 — 0t) — pe (03 + 01), (03 + 04) x' - (04 + 02) y' = c (05 - 0s) - Pe (0-s + ^)- If we express that 0 belongs to the enclosing four-system we shall have two linear equations to be also satisfied by tho co-ordinates of 0. These equations may be written without loss of generality in the form 0, = 0; 04 = 0. We have finally the equation Uo = 0 characteristic of the quadratic three- system. From these equations the co-ordinates are to be eliminated. But the eliminant of k equations in (k — 1) independent variables is a homo- geneous function of the coefficients of each equation whose order is, in general, equal to the product of the degrees of all the remaining equations*. In the present case, the coefficient of each of the first three equations must be of the second degree in the eliminant and hence, the resulting equation for pe is of the sixth degree, so that we have the following theorem. Of the screws which belong to a quadratic three-system, six can be drawn through any point. As the enclosing system in this case is of the fourth order, the screws of the enclosing system drawn through any point must lie on a cone of the second degree (§ 218). Hence it follows that the six screws just referred to must all lie on the surface of a cone of the second degree. We may verify the theorem just proved by the consideration that if the function Ue could be decomposed into two linear factors, each of those factors * Salmon, Modern Higher Algebra, p. 76, 4th Edition (1885).