194 Sir Robert Ball, Note on a [May 25, 



geometrical entities, are not affected by alterations in the mass. 

 This produces its effect in the twist velocity acquired and not in 

 the position of the screw. 



It thus appears that if the four screws had been chosen 

 arbitrarily we should have ten conditions to satisfy and only nine 

 disposable coordinates. It is hence plain that the four screws 

 cannot be chosen quite arbitrarily. They must be in some way 

 restricted. The object of the present note is to shew that these 

 restrictions are two in number and to set down what they are. 



Draw a cylindroid A through a, /3, and another cylindroid P 

 through 7], f. Then an impulsive wrench about any screw to on P 

 will make the body twist about some screw on A. As co moves 

 over P so will its correspondent travel over A. It is easy to 

 shew that any four screws on P will be equi-anharmonic with 

 their five correspondents on A, and that consequently the two 

 systems will be homographic. 



In general to establish the homography of two cylindroids 

 three corresponding pairs of screws must be given. And of course 

 there could be a triply infinite variety in the possible homo- 

 graphies. It is however a somewhat remarkable fact that in the 

 particular homography with which we are concerned there is no 

 arbitrary element. Given two 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 co be one unit then the twist velocity acquired by is 



cos (0co) 



~w 



where cos(0«) denotes the cosine of the angle between the two 

 screws 6 and co and where p e is the pitch of 6. If therefore p e be 

 zero then cos (Oco) must be zero. In other words, the two impulsive 

 screws co 1} co 2 on P which correspond to the two screws of zero 

 pitch 1 , do on A must be at right angles to them respectively. 

 This will at once identify the correspondents on P to two known 

 screws on A. 



We have thus determined two pans of correspondents, and we 

 can now find a third pair. For if co 3 be a screw on P reciprocal to 0. 2 

 then its correspondent 3 will be reciprocal to &> 2 . Thus we have 

 three pairs 6 1} 2> 3 on A, and their three correspondents co 1} co„, co 3 

 on P. This establishes the homography, and the correspondent 

 to any other screw co is assigned by the condition that the anhar- 

 monic ratio of w 1 , r«- 2 , &> 3 , <o is the same as that of 1} 0. 2 , 3 , 0. 



