A Treatise on the Theory of Screws

the theory or screws. [225 equated to zero would correspond to a three-system selected from the enclosing four-system. We know (§ 176) that three screws of a three-system can be drawn through each point. We have, consequently, three screws through the point for each of the two factors of Ue, i.e. six screws in all. . The equation of the 6th degree in pg contains also the co-ordinates x', y', z' in the sixth degree. Taking these as the current co-ordinates we may regard this equation as expressing the family of surfaces which, taken together, contain all the screws of the quadratic three-system. The screws of this system which have the same pitch pg are thus seen to be ranged on the generators of a ruled surface of the sixth degree. All these screws e ong of course to the enclosing four-system, and as they have the same pitches, they must all intersect the same pair of screws on the reciprocal cylindroid (§ 212). It follows that each of these pitch surfaces of the sixth degree must have inscribed upon it a pair of generators of the reciprocal cylindroid. Ascending one step higher in the order of the enclosing system we see that the quadratic four-system is composed of those screws whose co-ordinates satisfy one linear homogeneous equation Z = 0, and one homogeneous equation of the second degree U = 0. We may study these screws as follows. Let the direction cosines of a screw 0 be cos X, cos p, cos v. If the leference be made, as usual, to a set of canonical co-reciprocals we have cos X = 0X + 0„; cos p = 0S + 0t; cos v=05 + 0e, We therefore have for a point x', y', z' on 0 the equations (§ 218) = (a + pe) cos X — z' cos p + y' cos i>, 2a02 ~(a~ pg) cos X + z' cos p — y' cos v, with similar expressions for 03, 04> 0S! 0f._ Substituting these expressions in L = 0 and U = 0 and eliminating pe we obtain an homogeneous equation of the fourth degree in cos X,’ cos P, cos V. If we substitute for these quantities x — x', y — y', z — z', we obtain the equation of the cone of screws which can be drawn through x', y', z- this cone is accordingly of the fourth degree. We verify this con- clusion by noticing that if J7=0 were the product of two linear functions this cone would decompose into two cones of the second degree, as should clearly be the case (§ 218). It remains to consider the Quadratic Five-system. In this case the enclosing system includes every screw in space, and the six co-ordinates of