40.] SCREW MOTION. 21 



a single rotation about a certain axis passing through this point 

 to bring the whole body into its final position. It thus appears 

 that any displacement of a rigid body can be effected by sub- 

 jecting the body first to a translation and then to a rotation 

 (or vice versa, as is easily seen) ; and this can be done in an 

 infinite number of ways, as the displacement of any point of 

 the body may be selected for the translation. 



39. It is to be noticed that for all these different ways of 

 effecting the displacement M^M l the direction of the axis of 

 rotation and the angle of rotation are the same. To see this 

 more clearly, let the displacement be effected first by the trans- 

 lation A Q A 1 and a rotation of angle a about the axis a passing 

 through A l ; and then let the same displacement be produced 

 by the translation B^B^ of some other point B and a rotation qf 

 angle /3 about an axis b passing through B^. We wish to show 

 that a^ and b are parallel and that the angles a and ft are equal. 



Consider a plane TT of the rigid body which in its original 

 position TT O is perpendicular to the axis a v The translation 

 A Q A l transfers it into a parallel position and the rotation a about 

 a 1 turns it in itself into its final position TTJ ; hence TT O and TTJ 

 are parallel. The translation B B 1 likewise moves TT into a 

 position parallel to the original one ; and as its final position, 

 ir v is parallel to TT O , the axis of rotation b must necessarily be 

 perpendicular to TT O and 7r 1} that is b l must be parallel to a^. 



Again, any straight line / in TT remains parallel to its original 

 position / after the translations A A 1 and B^B V Its change of 

 direction is due to the rotations alone ; the angle of rotation 

 must therefore be the same for both rotations, viz. equal to the 

 angle (/Q/J) formed by the initial and final positions of the line /. 



40. Among the different combinations of a translation with 

 a rotation effecting the displacement M Q M l there is one of 

 particular importance ; it is that for which the axis of rotation 

 is parallel to the translation. 



Let us again consider the plane TT perpendicular to the com- 



