A Treatise on the Theory of Screws

226] FREEDOM OF THE FOURTH ORDER. 239 then the two roots of the quadratic are equal but with opposite signs, and hence (§ 119) we have the following theorem. If the condition 11^=0 is satisfied by the co-ordinates of two screws?? and £ which belong to the enclosing (». + l)-system, then these two screws y, t and the two screws which, lying on the cylindroid (r/, £), also belong to the quadratic «-system Ue = 0, will be parallel to the four rays of an harmonic pencil. We are now to develop the conception of polar screws alluded to in § 224, and this may be most conveniently done by generalizing from a well-known principle in geometry. Let 0 be a point and $ a quadric surface. Let any straight line through 0 cut the quadric in the two points X1 and X„. Take on this straight line a point P so that the section 0XP1X2 is harmonic; then for the different straight lines through 0 the locus of P is a plane. This plane is of course the well-known polar of P. We have an analogous conception in the present theory which appears as follows. Take any screw r/ in the enclosing (n + l)-system. Draw a pencil of n cylindroids through r/, all the screws of each cylindroid lying in the enclosing (n + l)-system. Each of these cylindroids will have on it two screws which belong to the quadratic n-systein Ue = 0. On each of these cylindroids a screw £ can be taken which is the harmonic conjugate with respect to r/ with reference to the two screws of the quadratic «-system which are found on the cylindroid. We thus have n screws of the f type, and these n screws will define an n-system which is of course included within the enclosing (n + l)-system. The equation of this n-system is obviously JT — Y I S' _ A This equation is analogous to the polar of a point with regard to a quadric surface. We have here within a given enclosing (n + l)-system a certain n-system which is the polar of a screw -7 with respect to a certain quadratic ?i-system. The conception of reciprocal screws enables us to take a further im- portant step which has no counterpart in the ordinary theory of poles and polars. The linear equation for the co-ordinates of £ namely ^ = 0, is merely the analytical expression of the fact that £ is reciprocal to the