293] DEVELOPMENTS OF THE DYNAMICAL THEORY. 311 



disposable quantities in the choice of rj, and five more in the choice of f. 

 We ought, therefore, to have ten disposable co-ordinates for the designing 

 and the placing of the rigid body. But there are not so many. We have 

 three for the co-ordinates of its centre of gravity, three for the direction of 

 its principal axes, and three more for the radii of gyration. The other 

 circumstances of the rigid body are of no account for our present purpose. 



It thus appears that if the four screws had been chosen arbitrarily we 

 should have ten conditions to satisfy, and only nine disposable co-ordinates. 

 It is hence plain that the four screws cannot be chosen quite arbitrarily. 

 They must be in some way restricted. We can show as follows that these 

 restrictions are not fewer than two. 



Draw a cylindroid A through a, fi, and another cylindroid P through rj. 

 Then an impulsive wrench about any screw &&amp;gt; on P will make the body 

 twist about some screw on A. As eo moves over P, so will its corre 

 spondent 6 travel over A. It is shown in 125 that any four screws on 

 P will be equianharmonic with their four correspondents on A, and that 

 consequently the two systems are homographic. 



In general, to establish the homography of two cylindroids, three cor 

 responding pairs of screws must be given ; and, of course, there could be 

 a triply infinite variety in the possible homographies. It is, however, a 

 somewhat remarkable fact that in the particular homography with which 

 we are concerned there is no arbitrary element. The fact that the rigid 

 body is supposed quite free distinguishes this special case from the more 

 general one of 290. Given the cylindroids A and P, then, without any 

 other considerations whatever, all the corresponding pairs are determined. 

 This is first to be proved. 



If the mass be one unit, and the intensity of the impulsive wrench on &&amp;gt; 

 be one unit, then the twist velocity acquired by 6 is ( 280) 



cos (#&&amp;gt;) 



~~^~ 



where cos(#o&amp;gt;) denotes the cosine of the angle between the two screws 

 and ft&amp;gt;, and where p g is the pitch of 6. If, therefore, p g be zero, then cos (#o&amp;gt;) 

 must be zero. In other words, the two impulsive screws co ly o&amp;gt; 2 on P, which 

 correspond to the two screws of zero pitch d l , 6 2 on A, must be at right 

 angles to them, respectively. This will in general identify the correspondents 

 on P to two known screws on A. 



We have thus ascertained two pairs of correspondents, and we can now 

 determine a third pair. For if o&amp;gt; 3 be a screw on P reciprocal to # 2 , then its 

 correspondent 3 will be reciprocal to o&amp;gt; 2 . Thus we have three pairs 6 lt 2 , # 3 

 on A, and their three correspondents co l , &&amp;gt; 2 , o&amp;gt;, on P. This establishes the 



