174] FREEDOM OF THE THIRD ORDER. 173 



pitches by p$ and p y . From the properties of the cylindroid it appears 

 that a, the semiaxis of the pitch quadric, must be determined from the 

 equations 



a = (pp py) sin I cos I, 



P cos 2 1 + p y sin 2 1 = ; 

 whence eliminating I, we deduce 



with of course similar values of b and c. Substituting these values in the 

 equation of the quadric 



we deduce the important result which may be thus stated : 



The three principal axes of the pitch quadric, when furnished with suitable 

 pitches p a ,pii, p-f, constitute screws belonging to the screw system of the third 

 order, and the equation of the pitch quadric has the form 



PO.CO&quot; + ppf + p y z 2 + p a pppy = 0. 



We can also show conversely that every screw 6 of zero pitch, which 

 belongs to the screw system of the third order, must be one of the generators 

 of the pitch quadric. For 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 9 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. 



