A Treatise on the Theory of Screws

225] FREEDOM OF THE FOURTH ORDER. 237 the screw 0 are subjected to no other relation than that implied by the quadratic relation Ue = 0. As before we may substitute for 0lt ...,06 from the equations 2a02 = (a + pø) cos X — z cos p + y' cos v, 2a02 = (a — pe) cos X + / cos p — y' cos v, with similar expressions for 2b0lt 2b0.2, &c. Introducing these values into ue = o, we obtain a result which may be written in the form + 2Bpe + (7 = 0, where A, B, C contain cos X, cos p, cos v in the second degree, and where a/, y', z enter linearly into B and in the second degree into C. Hence we see that on any straight line in space there will be in general two screws belonging to any quadratic five-system. For the straight line being given x', y', z' are given, and so are cos X, cos p, cos v. The equation just written gives two values for a pitch which will comply with the necessary conditions. If we consider pa and also x', y', z as given, and if we substitute for cos X, cos p, cos v the expressions x — y — y', z — z respectively, we obtain the equation of a cone of the second degree. Thus we learn that for each given pitch any point in space may be the vertex of a cone of the second degree such that the generators of the cone when they have received the given pitch are screws belonging to a given quadratic five-system. If the equation A0-B>~0 be satisfied, then the straight lines which satisfy this condition will be singular, inasmuch as each contains but a single screw belonging to the quadratic five-system. As cos X, cos p, cos v enter to the fourth degree into this equation it appears that each point in space is the vertex of a cone of the fourth degree, the generators of which when proper pitches are assigned to them will be singular screws of the quadratic five-system. If we regard cos X, cos p, cos v as given quantities in the equation AC - = 0, then this will represent a quadric surface inasmuch as x, y', z' enter to the second degree. This quadric is the locus of those singular screws of the quadratic five-system which are parallel to a given direction. Hence the equation must represent a cylinder.