DYNAMICS OF A RIGID BODY. 123 



can be found parallel to any given direction. All the 

 generators of the quadric are parallel to the cone 



(A -k}* + (p ft - k}f + (A - k] z- = o, 



and k can be determined so that this cone shall have one 

 generator parallel to the given direction; the quadric 

 can then be drawn, on which two generators will be 

 found parallel to the given direction ; one of these belongs 

 to S, while the other belongs to S'. 



It remains to be proved that each screw of S has 

 a pitch which is proportional to the inverse square of the 

 parallel diameter of the pitch quadric* 



Let r be the intercept on a generator of the cone 



(A - 

 by the pitch quadric 



A* 2 



then k = - 



but k is the pitch of the screw of S, which is parallel 

 to the line r. 



Nine constants ( 49) are required for the determina- 

 tion of a screw complex of the third order. This is the 

 same number as that required for the specification of a 

 quadric surface.f We hence infer, what is indeed other- 



*This theorem is connected with some purely geometrical theorems of 

 Plucher, who has shown (Neue Geometric des Raumes, p. 130) that k^x* 

 + k?y* + &z* + k\k<Jii = o, is the locus of lines common to three linear com- 

 plexes of the first degree. The axes of the three complexes are directed along 

 the co-ordinate axes, and the parameters of the complexes are k\ t 2 , 3 ; the 

 same author has also proved that the parameter of any complex belonging 

 to the (" dreigliedrigen Gruppe") is proportional to the inverse square of the 

 parallel diameter of the hyperboloid. 



t Salmon's Analytic Geometry of Three Dimensions, p. 35. 



