66.] 



ROTATIONS. 



35 



the plane /j/ 2 into the position 

 /' 2 , brings it into the position 

 /'/! which is parallel to the 

 original position /^ The 

 whole body has thus been 

 moved parallel to itself in the 

 direction L^L'^ and the mag- 

 nitude of this translation is 



\\ the following rotation, about 



Fig. 21. 



. 



sin-, 

 2 



(3) 



s 



where is the angle of rotation about each axis, and L 

 the distance of the axes. 



The order of the rotations is evidently not invertible. 



65. We have seen in the preceding article that two equal and 

 opposite rotations about parallel axes produce a translation at right 

 angles to the axes of rotation. A translation can therefore always 

 be replaced by two such rotations. It follows that a translation 



followed by a rotation about an axis at right angles to the direc- 

 tion of translation can be replaced by a single rotation about a 

 parallel axis. To find this resulting rotation it is only neces- 

 sary to replace the translation by two parallel equal and oppo- 

 site rotations having the same effect (Art. 64) ; the three 

 rotations so obtained have parallel axes and can therefore 

 (Art. 63) be combined into a single one. 



66. The case of two infinitely small rotations (Fig. 22) is 

 again of particular importance, as we shall see later on. The 



formulae of Arts. 59 and 63 

 become in this case 



k 



i 



Fig. 22. 



J^^- J J -L> J-^t f 



(I"') 



(2'") 



The axis / of the resulting rota- 

 tion lies therefore in the plane 



of the given axes l v / 2 and divides their distance in th< 



