A Treatise on the Theory of Screws

174] FREEDOM OF THE THIRD ORDER. 173 pitches by pß and py. From the properties of the cylindroid it appears that a, the semiaxis of the pitch quadric, must be determined from the equations a = (pß — py) sin I cos I, Pß cos21 +py sin21 = 0 ; whence eliminating I, we deduce a = ^J-pßPy, with of course similar values of b and c. Substituting these values in the equation of the quadric y'! z2 . ~2 4- ; 2 + -^ = 1 a2 (r c2 we deduce the important result which may be thus stated:— The three principal axes of the pitch quadric, when f urnished with suitable pitches pa,pß,py, constitute screws belonging to the screw system of the third order, and the equation of the pitch quadric has the form pa.!«' + Pßy" + PyZ2 + PaPßpy = 0. We can also show conversely that every screw 3 of zero pitch, which belongs to the screw system of the third order, must be one of the generators of the pitch quadric. For 3 must be reciprocal to all the screws of zero pitch on the reciprocal system of generators of the pitch quadric; and since two screws of zero pitch cannot be reciprocal unless they intersect either at a finite or infinite distance, it follows that 3 must intersect the pitch quadric in an infinite number of points, and must therefore be entirely contained thereon. 174. The Family of Quadrics. It has been shown that all the screws of given pitch belonging to a system of the third order are the generators of a certain hyperboloid. There is of course a different hyperboloid for each pitch. We have now to show that all these hyperboloids are concentric. Take any two screws whatever belonging to the system and draw the cylindroid which passes through those screws. This cylindroid contains two screws of every pitch. It must therefore have two generators in common with every hyperboloid of the family. But from the known sym- metrical arrangement of the screws of equal pitch on a cylindroid, it follows that the centre of that surface must lie at the middle point of the shortest distance between each two screws of equal pitch. The centres of the hyper- boloids for all possible pitches must therefore lie in the principal plane of any cylindroid of the system. Take any three cylindroids of the system.