CHAPTER II. 



THE CYLINDROID. 



9. Introduction. 



Let a and /3 be any two screws which we shall suppose to be fixed both 

 in position and in pitch. Let a body receive a twist of amplitude of about 

 a, followed by a twist of amplitude /3 about /3. The position attained 

 could have been arrived at by a single twist about some third screw p with 

 an amplitude p. We are always to remember that the amplitudes of the 

 twists are infinitely small quantities. With this assumption the order in 

 which the twists about a and /3 are imparted will be immaterial in so far 

 as the resulting displacements are concerned. The position attained is the 

 same whether a follows /3 or /3 follows a . 



Any change in a or in /3 will of course generally entail a change both in 

 the pitch and in the position of p. It might thus seem that p depended 

 upon two parameters, and that consequently the different positions of p 

 would form a doubly infinite series, known in linear geometry as a 

 congruence. But this is not the case, for we prove that p depends only 

 upon the ratio of a to /3 and is thus only singly infinite. 



Take any point P and let h a be the perpendicular distance from P to a, 

 while p a is as usual the pitch of the screw; then the point P is transferred 

 by the twist about a through the distance 



V^a 2 + / a 2 a . 

 The twist about /3 conveys P to a distance 



The resultant of these two displacements conveys P in a direction which 

 depends upon the ratio of a to /3 , and not upon their absolute magnitudes. 



Let P and Q be two points on p, then the resultant displacement will 

 convey P and Q to points P and Q respectively which are also on the axis 



