59-] 



ROTATIONS. 



single rotation or translation ; it will be shown in the next sec- 

 tion (Arts. 74-79) that they reduce to a twist, or screw motion. 



59. Intersecting Axes. The resultant of two successive rota- 

 tons, 0J about \ and # 2 about 1 2 , when the axes \ and 1 2 intersect 

 in a point O, is a single rotation of angle 6 about an axis 1 passing 

 through O. The trihedral formed by / x , / 2 and / has at / x a dihe- 

 dral angle = \ lt at / 2 a dihedral angle = \Q^ while its 

 exterior angle at /is =J#; that is, we have on a sphere of 

 radius I described about O : 



cos = 



cos 



sn 



sn 



cos 



(i) 



sn 



The truth of this proposition will appear by considering Fig.. 

 17. The rotation 6 1 about the axis / x brings the axis / 2 into its- 

 final position /' 2 . The rotation 0% 

 about l\ brings / x into its final 

 position l\. The planes bisecting 

 the dihedral angles O l at /j and 2 

 at l\ intersect in a line / which by 

 the rotation B 1 about l^ is brought ' 

 into the position I', and by the 

 rotation 2 about /' 2 is brought 

 back into its original position /. 

 The effect of the two rotations 

 taken in this order is therefore to 

 leave the line / in its place ; that 

 is, the resultant of the two succes- 

 sive rotations is a single rotation 

 about / as axis. Moreover, inspec- 

 tion of the figure shows that a 

 rotation about / by an angle equal 



to twice the exterior angle of the trihedral // x / 2 at / brings 

 and / 2 into their final positions l\ and l\. 



Fig. 17. 



