DYNAMICS OF A RIGID BODY. 1 65 



P, Q, R, S y T, [7, are so situated that forces acting 

 along them equilibrate when applied to a, free rigid body, 

 a certain determinant vanishes, and the six lines are 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 ,T MJ , y my z m , be a point on one of the lines, the direc- 

 tion cosines of the same line being a m , j3 m , 7, the condi- 

 tion is 



- y\<*\ 



o. 



02, p 2 , 72, J)Yy2-2 2 D 2 , 2 2 a 3 -# 2 7 2 , * 2 p 2 



tt 4 , fii, 74, J) ; 474 24)84, 2 4 a 4 - -#474? #4J34 

 5> HO, 7 5 y^li^ ~ ^of^S? 25(15 X*TJ5) ^5f^a 



A single screw JT must be capable of being found 

 which is reciprocal to all the six screws P, Q, R, S, T y U. 

 Suppose the rigid body were only free to twist about X, 

 then the six forces would not only collectively be in 

 equilibrium, but severally would be unable to stir the 

 body only free to twist about X. 



In general a body able to twist about six screws 

 (of any pitch) would have perfect freedom ; but the 

 body capable of rotating about each of the six lines, 

 Py Qy Ry Sy Ty Uy which B.rQ in involution, is not ne- 

 cessarily perfectly free (Mobius). 



* In the language of Pliicker (Neue Geometric des Raumes) a system of 

 lines in involution forms a linear complex. In our language a system of lines 

 in involution consists of the screws of equal pitch belonging to a screw complex 

 of the fifth orden and first degree. See also Salmon's Geometry of Three 

 Dimensions, third edition, p. 456, note. It may save the reader some trouble 

 to observe here that the word involution has been employed in a more gene- 

 ralised sense by Battaglini, and in quite a different sense by Klein. 



