GEOMETRY OF MOTION. 



[76. 



76. If the rotation dO and the translation ds are infinitesimal, 

 the axis of the resulting twist has (by Art. 68) a distance ds/dQ 

 from the axis / of the rotation dd and lies in the plane laid 

 through / at right angles to ds. 



77. Skew Axes. The resultant of two successive rotations 6 

 and 2 about two skew axes \ and 1 2 is a twist. This follows of 



course from the proposition of 

 Art. 40. The axis of the re- 

 sulting twist is the central axis \ 

 of the displacement ; its direc- 

 tion and position can be found 

 as in Art. 42. Fig. 26 illus- 

 trates the process. L^L^ is the 

 shortest distance of the axes 

 /!, / 2 . The first rotation, O v " 

 about /j, brings / 2 into its final 

 position /' 2 , an d L^ into L f 2 ', the 

 second rotation, 2 about /' 2 , 

 brings / x into its final position 

 /\, and L into L\. The axis 

 / of the resulting twist will 

 evidently be the shortest dis- 

 tance of the bisectors of the angles Z 2 1 Z' 2 and L^L\L\. 

 For a rotation about this line / brings / 2 into /' 2 and / x into l\. 



78. The angle of the resulting twist is the same as the angle 

 of the rotation resulting from two rotations V # 2 about two 

 intersecting axes parallel to the given axes / 1? / 2 . For (by Art. 

 65) either one of the rotations, say 2 about / 2 , may be replaced 

 by a rotation of the same angle 2 about an axis parallel to / 3 

 and intersecting l lt combined with a translation at right angles 

 to / 2 . The two rotations about the intersecting axes can then 

 be combined into a single rotation, and the angle and direction 

 of the axis of this latter rotation are not changed by combi- 

 nation with the translation (Art. 74). 



Fig. 26. 



