750 TWISTS. 39 



Thus it is seen that the combination of an infinitely small 

 rotation dO, with an infinitely small translation ds at right angles 

 to the axis of rotation, produces a single rotation of the same 

 angle about a parallel axis at a distance ds/d# from the original 

 axis in the plane through this axis perpendicular to the direction 

 of translation. 



73. Exercises. 



(1) The telescope of a theodolite, originally horizontal and pointing 

 north, is tipped into an elevation of 60, and then turned into the prime 

 vertical so as to point west. What single rotation is equivalent to the 

 two successive rotations? 



(2) In the preceding example, what would be the result of inverting 

 the order of the two rotations ? 



(3) The motion of a man in walking may be approximately described 

 as consisting at every step of two rotations of the body about parallel 

 axes perpendicular to the direction of motion, one axis passing through 

 the hip-joint, the other through the foot that remains on the ground 

 while the other foot is thrown forward. Find the angle of swing 

 (assuming the two rotations to be equal and opposite) if the length 

 of the step is 15 inches and the height of the hip-joint 3^ feet. 



3. SCREW MOTIONS ; TWISTS. 



74. We have seen in Arts. 40, 41 that a twist, i.e. a rota- 

 tion combined with a translation parallel to the axis of rotation, 

 constitutes the most general form of the displacement of a 

 rigid body. We proceed to discuss the most important cases 

 of the compositions of rotations and translations resulting in 

 twists. 



75. A rotation of angle 6 about an axis / can be combined 

 with a translation whose vector is s, by resolving s into two 

 components ; s l perpendicular to /, and s 2 parallel to /. The 

 former component combines (by Art. 65) with the rotation into 

 a single rotation of the same angle 6 about an axis parallel to /. 

 The result is therefore a rotation accompanied by a translation 

 s% parallel to the axis of rotation, i.e. a twist. 



