150 DYNAMICS OF A RIGID BODY. 



the second and third order ( 42, 112), so now we see- 

 that the same must be likewise true for the fourth, fifth,, 

 and sixth orders. 



The actual value of this constant for any given screw- 

 complex is evidently a characteristic feature of that 

 screw complex. 



137. Special Screw of the Complex. In general there- 

 is one line in each csrew complex of the fourth order, 

 which forms a screw belonging to the screw complex, 

 whatever be the pitch assigned to it. The line in ques- 

 tion is the nodal line of the reciprocal cylindroid. The 

 kinematical statement is as follows : 



When a rigid body has freedom of the fourth order, 

 there is in general one straight line, about which the body 

 can be rotated, and parallel to which it can be translated. 



138. Particular Case. A body which has freedom of 

 the fourth order may be illustrated by the case of a rigid 

 body, one point P of which is constrained to a certain 

 curve. The position of the body will then be specified 

 by four quantities, viz., the arc of the curve from a fixed 

 origin up to P, and three rotations about three axes 

 intersecting in P. The reciprocal cylindroid will in 

 this case assume an extreme form; it consists of screws 

 of zero pitch on all the normals to the curve at P. 



139. Statics. When a rigid body has freedom of 

 the fourth order, the necessary and sufficient condition 

 for equilibrium is, that the forces shall constitute a 

 wrench upon a screw of the cylindroid reciprocal to the 

 given screw complex. Thus, if one force can act on the 

 body without disturbing equilibrium, then this force 

 must lie on one of the two screws of zero pitch on the 

 cylindroid. If there were no real screws of zero pitch 

 on the cylindroid that is, if the pitch conic were an 

 ellipse, then it is impossible for equilibrium to subsist 

 when a force acts. It is, however, worthy of remark,. 



