A Treatise on the Theory of Screws

112 THE THEORY OF SCREWS. [123- second plane is that drawn through the point, and through the other screw on the cylindroid, of equal pitch to that which passes through the point. We have, therefore, solved in the most general manner the problem of the equilibrium of a rigid body with two degrees of freedom. We have shown that the necessary and sufficient condition is, that the resultant wrench be about a screw reciprocal to the cylindroid expressing the freedom, and we have seen the manner in which the reciprocal screws are distributed through space. We now add a few particular cases. 124. Particular Cases. A body which has two degrees of freedom is in equilibrium under the action of a force, whenever the line of action of the force intersects both the screws of zero pitch upon the cylindroid. If a body acted upon by gravity have freedom of the second order, the necessary and sufficient condition of equilibrium is, that the vertical through the centre of inertia shall intersect both of the screws of zero pitch. A body which has freedom of the second order will be in equilibrium, notwithstanding the action of a couple, provided the axis of the couple be parallel to the nodal line of the cylindroid. A body which has freedom of the second order will remain in equilibrium, notwithstanding the action of a wrench about a screw of any pitch on the nodal line of the cylindroid. 125. The Impulsive Cylindroid and the Instantaneous Cylin- droid. A rigid body M is at rest in a position P, from which it is either partially or entirely free to move. If M receive an impulsive wrench about a screw Xlt it will commence to twist about an instantaneous screw Alt if, however, the impulsive wrench had been about X., or X3 (M being in either case at rest in the position P) the instantaneous screw would have been A,, or A... Then we have the following theorem:— If Xlt X2, X3 lie upon a cylindroid $ (which we may call the impulsive cylindroid), then _42, A:) lie on a cylindroid S' (which we may call the instantaneous cylindroid). For if the three wrenches have suitable intensities they may equilibrate, since they are co-cylindroidal; when this is the case the three instantaneous twist velocities must, of course, neutralise; but this is only possible if the instantaneous screws be co-cylindroidal (§ 93).