202 VI. SYSTEMS OF VECTORS. DISTRIBUT. OF MASS. INSTANT. MOTION. 



If the rotations c^ and o 2 are of opposite signs and of equal 

 magnitudes, the intersection of the two bisectors is at infinity and 

 the axis of rotation is thus at infinity. A motion about an infinitely 

 distant axis is a translation. The direct proof is as follows. 



Let A be the center of rotation a , bringing B to B'. Then 

 rotate about B' through an equal angle in the opposite direction, 



bringing A to A. Triangles 

 ABB' and AAB' have AB' 

 common, and AB = A'B' and 

 the included angles equal, 

 therefore A A' and BB' are 

 equal and parallel and two 

 points consequently all 



points have moved parallel 



to each other the same distance. The motion is therefore a trans- 

 lation of magnitude, 



4) 



. 43. 



t = 2AB sin ~ 



Accordingly every translation may be decomposed into rotations, and we 

 may reduce all displacements to rotations. 



57. Rotations about Intersecting Axes. Infinitesimal 

 Rotations. Let OA and OB be two intersecting axes about which 



we revolve the body through the 

 angles c^ and o> 2 respectively. 

 Describe a sphere with the center 0. 

 Let the rotation o 1 about A bring 

 B to B f , and o> 2 about B' bring 

 A to A. Pass planes through 

 the vertices bisecting the angles oc^ 

 and 2 , then, as in 56, the 

 displacement just given is equi- 

 valent to a rotation about the line 

 of intersection CO of these planes. 

 The order of the rotations affects 

 the result. 



Since AC bisects the angle 

 BAB' and the spherical triangle 



Fig. 44. 



BAB' is isosceles, 



angle ABC = angle AB' C = y- 



Thus the resultant rotation, o = angle ACA' = angle BCB'. 

 Angle ACE= angle B 1 CD = angleDOJ5= |- 



