124 DYNAMICS OF A RIGID BODY. 



wise manifest, viz., that when the pitch quadric is 

 known the entire screw complex of the third order is 

 determined. 



Another very interesting property of the pitch qua- 

 dric is thus enunciated. Any three co-reciprocal screws 

 of a given screw complex of the third order are parallel to' a 

 triad of conjugate diameters of its pitch quadric. 



Take any three co-reciprocal screws of the complex 

 as screws of reference, and let/!, / 2 , /a be their pitches. 

 If then the co-ordinates of any screw p belonging to the 

 complex be denoted by p ly p 2 , p 3 , we shall have for the 

 pitch of p (65) 



A>=/1P1 2 +AP2 2 +/3P3 2 . 



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 ( 37) 



x y z 



~ = PI, r = ? 2 , - = ? 3 . 



It follows that pp must be proportional to the inverse 

 square of the parallel diameter of the quadric surface 



But p 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 diame- 

 ters, and hence three conjugate diameters of the pitch 

 quadric are parallel to three co-reciprocal screws of the 

 screw complex. 



We see from this that the sum of the reciprocals of 



