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 

 X lt it will commence to twist about an instantaneous screw A 1} if, however, 

 the impulsive wrench had been about X 2 or X 3 (M being in either case at 

 rest in the position P) the instantaneous screw would have been A 2 , or A 3 . 

 Then we have the following theorem: 



If Xi, X z , X 3 lie upon a cylindroid S (which we may call the impulsive 

 cylindroid), then A 1} A.,, A 3 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). 



