A Treatise on the Theory of Screws

 then 177 176] FREEDOM OF THE THIRD ORDER. Let r be the intercept on a generator of the cone (pa -k)x2 + (pß - k) f - lc) z* = 0; by the pitch quadric P°& + Pfrf + PA? + PaPßPy = 0 ; Je _ _PAP&Py. ri ’ but k is the pitch of the screw of S, which is parallel to the line r. Nine constants (§75) are required for the determination of a screw system of the third order. This is the same number as that required for the specification of a quadric surface. We hence infer, what is indeed otherwise manifest, viz., that when the pitch quadric is known the entire screw system of the third order is determined. Another interesting property of the pitch quadric is thus enunciated. Any three co-reciprocal screws of a given screw system of the third order are parallel to a triad of conjugate diameters of its pitch quadric. Take any three co-reciprocal screws of the system as screws of reference, and let plt p.,, p3 be their pitches. If then the co-ordinates of any screw p belonging to the system be denoted by P1, p2, p3, we shall have for the pitch of p (§ 95) 1 Ps = PA7 + pp7 + p-.pl If a parallelepiped be constructed, of which the three lines parallel to the reciprocal screws, drawn through the centre of the pitch quadric, are conterminous edges, and of which the line parallel to p is the diagonal, and if x, y, z be the lengths of the edges, and r the length of the diagonal, then we have (§ 35) x y z r = Pi’ r = P?> ~ = Pi- ll follows that pp must be proportional to the inverse square of the parallel diameter of the quadric surface pX ApAf+pA? = H. But p? must be proportional to the inverse square of the parallel diameter of the pitch quadric, and hence the equation last written must actually be the equation of the pitch quadric, when H is properly chosen. But the equation is obviously referred to three conjugate diameters, and hence three conjugate diameters of the pitch quadric are parallel to three co-reciprocal screws of the screw system. B. 12