212, 213] FREEDOM OF THE FOURTH ORDER. 219 



tangent from the point must be the line at infinity itself, and as the line 

 at infinity touches the conic, the envelope must be a parabola. 



In general there is one line in each screw system of the fourth order, 

 which forms a screw belonging to the screw system, whatever be the pitch 

 assigned to it. The line in question is the nodal line of the cylindroid 

 reciprocal to the four-system. 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. 



A body which has freedom of the fourth order may be illustrated by the 

 particular case where one point P of the body is forbidden to depart from 

 a given curve. The position of the body will then be specified by four 

 quantities, which may be, for example, 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 has de 

 generated to a plane, and in fact consists of screws of zero pitch on all the 

 normals to the curve at P. 



It is required to determine the locus of screws parallel to a given straight 

 line L, and belonging to a screw system of the fourth order. The problem 

 is easily solved from the principle that each screw of the screw system must 

 intersect at right angles a screw of the reciprocal cylindroid ( 22). Take, 

 therefore, that one screw 6 on the cylindroid which is perpendicular to L. 

 Then a plane through 6 parallel to L is the required locus. 



213. Equilibrium with freedom of the Fourth Order. 



When a rigid body has freedom of the fourth order, it is both necessary 

 and sufficient for equilibrium, that the forces shall constitute a wrench upon 

 a screw of the cylindroid reciprocal to the given screw system. Thus, if a 

 single 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 would be impossible for equilibrium to 

 subsist under the operation of a single force. It is, however, worthy of 

 remark, that if one force could act without disturbing the equilibrium, 

 then in general another force (on the other screw of zero pitch) could 

 also act without disturbing equilibrium. 



A couple which is in a plane perpendicular to the nodal line can be 

 neutralized by the reaction of the constraints, and is, therefore, consistent 

 with equilibrium. In no other case, however, can a body which has freedom 

 of the fourth order be in equilibrium under the influence of a couple. 



