DYNAMICS OF A RIGID BODY. 103 



twists about /3 and a must move the point in the same 

 direction. 



96. Properties of the Pitch Conic. Since the pitch of 

 a screw on a cylindroid is proportional to the inverse 

 square of the parallel diameter of the pitch conic ( 20), 

 the asymptotes must be parallel to the screws of zero 

 pitch ; also since a pair of reciprocal screws are paral- 

 lel to a pair of conjugate diameters ( 42), it follows that 

 the two screws of zero pitch, and any pair of reciprocal 

 screws, are parallel to the rays of an harmonic pencil.* 

 If the pitch conic be an ellipse, there are no real screws 

 of zero pitch. If the pitch conic be a parabola, there is 

 but one screw [of zero pitch, and this must be one of 

 the two screws which intersect at right angles. Since 

 this screw is reciprocal to itself, as well as to the screw 

 it intersects, it must be reciprocal to every screw on the 

 cylindroid ( 24). This is the only case in which a screw 

 on the cylindroid is reciprocal to the cylindroid. 



97. Equilibrium of a Body with Freedom of the Second 

 Order. We shall now consider more fully the conditions 

 under which a body which has freedom of the second 

 order is in equilibrium. The necessary and sufficient 

 condition is, that the forces which act upon the body 

 shall constitute a wrench upon a screw which is reci- 

 procal to the cylindroid which defines the freedom of the 

 body. 



It has been shown ( 25), that the screws which are 

 reciprocal to a cylindroid exist in such profusion, that 

 through every point in space a cone of the second order 

 can be drawn, of which the entire superficies is made up 

 of such screws. We shall now examine the distribution 

 of pitch upon such a cone. 



* Salmon's Conic Sections, 3rd Edition, p. 273. 



