A Treatise on the Theory of Screws

174 THE THEORY OF SCREWS. [174, The centres of all the hyperboloids coincide with the intersection of the three principal planes of the cylindroids. It will be convenient to call this point the centre of the three-system. We hence see that whenever three screws of a three-system are given, the centre of the system is determined as the intersection of the principal planes of the three cylindroids defined by each pair of screws taken suc- cessively. We may also show that not only are the family of hyperboloids concentric, but that they have also their three principal axes coincident in direction and situation with the principal axes of the pitch quadric. Draw any principal axis z of the pitch quadric. Two screws of zero pitch belonging to the system will be intersected by z and we draw the cylindroid through these two screws. Let and L., be the two screws of equal pitch p on this cylindroid. Let 0 be the centre of the cylindroid, this same point being also the centre of the pitch quadric, and therefore as shown above of every p-pitch hyperboloid Sp. As the centre bisects every diameter, it follows that the plane 0L2 cuts the hyperboloid Sp again in a ray If which is perpendicular to z and crosses L, at its intersection with z. The plane containing L, and £/ is therefore a tangent to Sp at the point where the plane is cut by z. As z is perpendicular to this plane it follows that the diameter is perpendicular to its conjugate plane. Hence z is a principal axis of ßp, and the required theorem is proved. Let now >8 denote a screw system of the third order, where a, ß, y are the three screws of the system on the principal axes of the pitch quadric. Diminish the pitches of all the screws of 6' by any magnitude k. Then the quadric (pa — k) x2 + (pß -k)y2 + (py -k')z2 + (pa - k)(pß -k)(py- /c) = 0... must be the locus of screws of zero pitch in the altered system, and therefore of pitch + k in the original system (§ 110). Regarding k as a variable parameter, the equation just written represents the family of quadrics which constitute the screw system $ and the reciprocal screw system S'. Thus all the generators of one system on each quadric, with pitch + k, constitute screws about which the body, with three degrees of freedom, can be twisted; while all the generators of the other system, with pitch — k, constitute screws, wrenches about which would be neutralized by the reaction of the constraints. For the quadric to be a real surface it is plain that k must be greater than the least, and less than the greatest of the three quantities pa, pß, py.