444 Proceedings of the Royal Irish Academ?/. 



which the body can accept is limited, and the body can, indeed, only 

 twist about one of the singly infinite number of screws which consti- 

 tute a cylindroid. To any screw in space will correspond one screw 

 on the cylindroid. But will it be correct to say, that to one screw on the- 

 cylindroid corresponds one screw in space ? The fact is, that there- 

 are a quadruply infinite number of screws, an impulsive wrench on 

 any one of which will make the body choose the same screw on the 

 cylindroid for its instantaneous movement. The relation of this qua- 

 druply infinite group is well known in the Theory/ of Screivs. It is 

 shown {Screivs, p. 110) that, given a screw a on the cylindroid, there 

 is one screw 6 on the cylindroid, an impulsive wrench on which will 

 make the body commence to twist about a. It is further shown that 

 any screw wliatever which fulfils the single condition of being reci- 

 procal to a single specified screw on the cylindroid possesses the same 

 property. The screws thus form a 5-system. The correspondence 

 at present before us is therefore to be thus stated — 



To one screw in space corresponds one screw on the cylindroid, mid to 

 one screiv on the cylindroid corresponds a 5-system in space. 



We may look at the matter in a more general manner. Consider 

 an m-system [A) of screws, and an ?^-system {B) {m>n). If we make 

 7n = 6 and n = 2, this system includes the system we have been just 

 discussing. To one screw in A will correspond one screw in B, but 

 to one screw in B will correspond, not a single screw in A, but an 

 (m + 1 — n) system of screws. 



'Li m = n, we find that one screw of one system corresponds to one 

 screw of the other system. Thus, if m - n = 2, we have a pair of 

 cylindroids, and one screw on one cylindroid corresponds to one screw 

 on the other. A set of four screws on a cylindroid, being all parallel 

 to a plane, we may speak of the anharmonic ratio of four co-cylin- 

 droidal screws, and we obtain the result that it is equal to the anhar- 

 monic ratio of the four corresponding screws {Screws, p. 106). If 

 m = 3, and n = 2, we see that to each screw on the cylindroid will cor- 

 respond a whole cylindroid of screws belonging to the 3-system. For 

 example, if a body have freedom of the second order, a whole cylin- 

 droid full of screws can always be chosen from any 3-system, an im- 

 pulsive wrench on any one of which Avill make the body commence to 

 twist about the given screw. 



The property of the screws common to the two homographic 

 systems will of course require some modification when we are only 

 considering an m-system and an w-system. Let us take the case of a 

 3-8ystem on the one hand, and a 6-system, or all the screws in space, 

 on the other hand. To each screw a of the S-system A must corre- 

 spond, a 4-system, B in space. The screws of this 4-system are in 

 such profusion, that a whole cone full of them can be di-awn through 

 every point in space. Amid this multitude it is most interesting to 

 note that one screw jS can be found, which, besides belonging to B, 

 belongs also to A. Take any two screws reciprocal to B, and any 



