of a Free or Constrained Rigid Body, 211 



A body which has two degrees of freedom is therefore able 

 to twist about all the screws which lie upon a cylindroid. This 

 is true, no matter what be the nature of the constraints. It 

 is a matter worthy of notice that notwithstanding the infinite 

 variety of constraints which would permit a body to have two 

 degrees of freedom, that freedom must still be completely de- 

 fined by a cylindroid, although all cylindroids are similar sur- 

 faces, and possess no variety except as to absolute size. It 

 should also be remarked that the pitches of the screws on the 

 cylindroid are proportional to the inverse squares of the paral- 

 lel diameters of a certain conic. 



If it be found that a body can be twisted about three screws 

 which do not lie upon the same cylindroid, then the body must 

 be capable of being twisted about a doubly infinite number of 

 screws. Of this doubly infinite number three pass through 

 each point in space, all the screws of given pitch lie upon a 

 hyperboloid, and all the screws parallel to a plane lie upon a 

 cylindroid. The pitch of each screw of the system is propor- 

 tional to the inverse square of the parallel diameter of a certain 

 quadric called the pitch-quadric ; and the pitch-quadric is 

 itself the locus of the screws of zero pitch. The entire sys- 

 tem is determined when the pitch-quadric is known. These 

 kinematical theorems are intimately connected with Pliicker's 

 geometrical speculations on a system of three linear complexes. 



Included in the case of freedom of the third order, we have 

 the celebrated problem where a body is rotating around a 

 point. In this case, however, the pitch-quadric assumes an 

 evanescent form, and the general conception of the capabilities 

 of a body which has freedom of the third order are very much 

 degraded. 



If a body be able to twist about four screws which do not 

 all belong to such a system as that we have just been describing, 

 then the body must be able to twist about a trebly infinite 

 number of screws. A cone of screws of this system passes 

 through each point of space ; and the cone may be drawn by 

 letting fall perpendiculars from the point upon the generators 

 of a cylindroid. It is remarkable that these cones are of the 

 second order ; and it can also be shown that the feet of their 

 perpendiculars upon the generators of the cylindroid are in 

 the same plane. It is thus to be observed that the capabilities 

 of motion possessed by a body which has freedom of the fourth 

 order are completely determined when a certain cylindroid is 

 given in size and position ; for then all the cones are deter- 

 mined. 



In the case where a body can twist about five screws 

 not belonging to a system such as that we have just been 



