1 64 DYNAMICS OF A RIGID BODY. 



sacrificed no generality in the assumption that 7"passed 

 through the origin. 



Since the sexiant is linear in x^y^ z ly it appears that 

 all parallel screws of given pitch reciprocal to one screw 

 lie in a plane. Since the sexiant is linear in a : , /3i, ji 

 we have Mobius' theorem ( 80). 



The property possessed by six screws when their 

 sexiant vanishes may be enunciated in different ways,, 

 which are precisely equivalent. 



(a). The six screws are all reciprocal to one screw. 



(b}. The six screws are members of a screw complex 

 of the fifth order and first degree. 



(c}. Wrenches of appropriate intensities on the six 

 screws equilibrate, when applied to a free rigid body. 



(d). Properly selected twist velocities about the six 

 screws neutralize, when applied to a rigid body. 



(e). A body might receive six small twists about the- 

 six screws, so that after the last twist the body would 

 occupy the same position which it had before the first. 



If seven wrenches equilibrate (or twists neutralize), 

 then the intensity of each wrench (or the amplitude of 

 each twist) is proportional to the sexiant of the six non- 

 corresponding screws. 



153. Equilibrium. For a rigid body which has free- 

 dom of the fifth order to be in equilibrium, the necessary 

 and sufficient condition is that the forces which act upon 

 the body constitute a wrench upon that one screw ta 

 which the freedom is reciprocal. We thus see that it is 

 not possible for a body which has freedom of the fifth 

 order to be in equilibrium under the action of gravity 

 unless the screw reciprocal to the freedom have zero 

 pitch, and coincide in position with the vertical through 

 the centre of inertia. 



Professor Sylvester has shown* that when six lines, 



* Comptes Rendus, tome 52, p. 816. See also p. 741. 



