DYNAMICS OF A RIGID BODY. 1 5 i 



that if one force can act without disturbing the equili- 

 brium, then another force (on the other screw of zero 

 pitch) will be in the same predicament. 



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 equili- 

 brium. In no other case, however, can a body which 

 has freedom of the fourth order be in equilibrium under 

 the influence of a couple. 



140. Equilibrium of Five Forces. The five forces must, 

 if the body be free, belong to a screw complex of the 

 fourth order. Draw the cylindroid reciprocal to the com- 

 plex. The five forces must, therefore, intersect both the 

 screws of zero pitch on the cylindroid. We, therefore, 

 have the well-known condition that two straight lines 

 can be drawn which intersect all the five forces. Four 

 of the forces will determine the two lines, and therefore 

 the fifth force may enjoy any liberty consistent with the 

 requirement that it also intersects the two lines. This 

 condition is also a sufficient one, so far as the positions 

 of the forces are concerned. 



If A j, ... A 5 be the five forces, the ratio of A l : A* is 

 thus determined. 



Let P, Q be the two screws of zero pitch upon the 

 cylindroid. 



Let X, Ybe two screws reciprocal to'^, A z . 



Let Z be a screw reciprocal to A 3 , A^ A 5 . 



Construct the screw / reciprocal to the five screws 



X, Y, P, Q, Z. 



Now, the four screws X, Y, P, Q are reciprocal to 

 the cylindroid A l9 A z ; therefore 7, which is reciprocal 

 to X, Y y R y P y Q, must lie upon the cylindroid (Ai,A t ). 



Since P, Q, Z are all reciprocal to A z , A^ A s , it fol- 

 lows that / being reciprocal to P, Q, Z must belong to 



