CHAPTER X. 



THE DYNAMICS OF A RIGID BODY WHICH HAS FREEDOM 

 OF THE SECOND ORDER. 



86. The Screw Complex of the Second Order. When a 

 rigid body is capable of being twisted about two screws 

 and 0, it is capable of being twisted about every screw 

 on the cylindroid (0, 0). If it also appear that the body 

 cannot be twisted about any screw which does not lie 

 on the cylindroid, then the body is said to have freedom 

 of the second order, and the cylindroid is the screw com- 

 plex of the second order by which the freedom is de- 

 fined. 



Eight numerical data are required for the determina- 

 tion of a cylindroid. We must have four for the specifi- 

 cation of the nodal line, two more are required to define 

 the extreme points in which the surface cuts the nodal 

 line, one to assign the direction of one generator, and 

 one to give the pitch of one screw, or the eccentricity of 

 the pitch conic. 



Although only eight constants are required to define 

 the cylindroid, yet ten constants must be used in de- 

 fining two screws 0, 0, from which the cylindroid could 

 be constructed. The ten constants not only define the 

 cylindroid, but also point out two special screws upon 

 the surface. 



87. Applications of Screw Co-ordinates. We have 

 shown (42) that if a, )3 be the two screws of a cylin- 

 droid, which intersect at right angles, that then the 

 co-ordinates of any screw 0, which makes an angle / 

 with the screw a, are : 



