176] FREEDOM OF THE THIRD ORDER. 177 



Let r be the intercept on a generator of the cone 



(p a - k) tf + (pp -k)f + (p y -k)z* = 0; 



by the pitch quadric 



p a x&quot; + ptf + W&amp;gt; + paptpy = ; 



then k = -PP^- 



r- 



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 p lt p^, p 3 be their pitches. If then the co-ordinates of any screw p 

 belonging to the system be denoted by p l} p 2 , p 3 , we shall have for the pitch 

 ofp(95) 



Pp = PlP* + P*P* 



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 



~ = Pi&amp;gt; - = P* -=P*- 



It follows that p p must be proportional to the inverse square of the 

 parallel diameter of the quadric surface 



P& 2 + p 2 y n - + p 3 z = H. 



But p ft 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 



