DYNAMICS OF A RIGID BODY. IOI 



From this we see that three of the pitches of a set of 

 six co-reciprocals must be +, and three must be -.* For, 

 suppose that the pitches of four of the co-reciprocals had 

 the same sign, and let ij be a screw perpendicular to the 

 two remaining co-reciprocals, then the identity just writ- 

 ten would reduce to the sum of four positive terms equal 

 zero, which is absurd. 



94. Screws on One Line. There is one case in which 

 a body has freedom of the second order that demands 

 special attention. Suppose the two given screws 0, <p, 

 about which the body can be twisted, happen to lie on 

 the same straight line, then the cylindroid becomes illu- 

 sory. If the amplitudes of the two twists be &, 0', then 

 the body will have received a rotation & + 0', accom- 

 panied by a translation Q f p 9 + fy'p$. This movement is 

 really identical with a twist on a screw of which the 

 pitch is : 



Since ^, 0' may have any ratio, we see that, under these 

 circumstances, the screw complex which defines the 

 freedom consists of all the screws with pitches ranging 

 from - co to + oo, which lie along the given line. It fol- 

 lows ( 93), that the co-ordinates of all the screws about 

 which the body can be twisted are to be found by giving 

 p.$ all the values from - oo to + oo in the expressions : 



4/6 



* This interesting theorem was communicated to me by Dr. Klein, who had 

 proved it as a property of the parameters of " six fundamental complexes in in- 

 volution" (Math. Ann. Band, ii., p. 204). 



