A Treatise on the Theory of Screws

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 l)c rotated, tind 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. Ihe 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). lake, therefore, that one screw d on the cylindroid which is perpendicular to L. Then a plane through d 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.