CHAPTER XIII. 



THE GEOMETRY OF THE CYLINDROID. 



159. Another investigation of the Cylindroid. 



The laws of the composition of twists and wrenches are of such funda 

 mental importance in the present subject that the following method* of 

 investigating the cylindroid seems worthy of attention. This method is not, 

 however, presented as a substitute for that already given (Chap. II.) which is 

 certainly both more simple and more direct. 



Let a and /3 be any two screws, then if a body receives a twist about a, 

 followed by another twist about /3, the position arrived at could have been 

 reached by a single twist about a third screw 7. If the amplitudes of the 

 twists about a and ft are given, then the position of 7, as well as the ampli 

 tude of the resultant twist thereon, are, of course, both determined. If, 

 however, the amplitudes of the twists on a and ft are made to vary while 

 the screws a and ft themselves remain fixed, then the position of 7, no less 

 than the amplitude of the resultant twist, must both vary. It has however 

 been shown in 9 that the position and pitch of 7 remain constant so long as 

 the ratio of the amplitudes of the twists about a and ft remains unchanged. 

 As this ratio varies, the position of 7 will vary, so that this position is a 

 function of a single parameter ; and, accordingly, 7 must be restricted to be 

 one of the generators of a certain ruled surface S, which includes a and ft as 

 extreme cases in which the ratio is zero and infinity respectively. 



Let 6 l be a screw which is reciprocal both to a and ft, then 6 l must also 

 be reciprocal to every screw 7 on S. Let 2 , 3&amp;gt; # 4 be three other screws also 

 reciprocal to S. Since a screw is defined by five conditions, it is plain that a 

 screw which fulfils the four conditions of being reciprocal to lf # 2 &amp;gt; 0s&amp;gt; $4 will 

 have one disposable parameter, and must, therefore, be generally confined to 

 a certain ruled surface. This surface must include S, inasmuch as all the 

 screws on S are reciprocal to l} 2 , 3 , 4 ; but further, it cannot include any 



* Proceedings of the Royal Irish Academy, 2nd Ser., Vol. iv. p. 518 (1885). 



