230] 



FREEDOM OF THE FIFTH ORDER. 



249 



(6) The six screws are members of a screw-system 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. 



For a rigid body which has freedom 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 to 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. 



Sylvester has shown* that when six lines, P, Q, R, S, T, U, are so situated 

 that forces acting along them equilibrate when applied to a free rigid body, 

 a certain determinant vanishes, and he speaks of the six lines so related as 

 being in involution^. 



Using the ideas and language of the Theory of Screws, this determinant 

 is the sexiant of the six screws, the pitches of course being zero. 



If x m , y m , z m , be a point on one of the lines, the direction cosines of the 

 same line being a m , /3 m ,y m , the condition is 



ii-#i7i&amp;gt; #i/3i-2/ii =0. 



, 72, y-fli- 



/3 3 , 73, y 3 y 3 - 



&, 74, 2/474 - 



&, 75, y 5 %- 



- #373, 



~ 2/44 



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



t In our language a system of lines thus related consists of the screws of equal pitch belonging 

 to a five-system. In the language of Pliicker (Neue Geometric des Raumes) a system of lines 

 in involution forms a linear complex. It may save the reader some trouble to observe here 

 that the word involution has been employed in a more generalised sense by Battaglini, and in 

 quite a different sense by Klein. 



