KINEMATICS. [294. 



294. If a rigid body be subjected at the time t to two simul- 

 taneous angular velocities co v &> 2 about skew (or crossing, i.e. 

 not intersecting and not parallel) axes / x , / 2 , or if it be subjected 

 to an angular velocity co about an axis / and a simultaneous 

 linear velocity v not perpendicular to /, its state of motion 

 during the time dt cannot be expressed by a single angular or 

 linear velocity. 



The body can be said to have in either case a twist-, or screw- 

 velocity, i.e. an angular velocity co about an axis / combined with 

 a linear velocity z/ parallel to this axis. 



To prove this in the latter of the two cases it is only necessary 

 to resolve v into a component z/ parallel to / and a component 

 v' perpendicular to /. The latter, being equivalent to a rotor 

 couple (&>, &>) of moment v' =pco (see Art. 256), combines with 

 the given angular velocity co about / into an angular velocity co 

 about a parallel axis /' at the distance p = v' /a> from /. The 

 combination of the angular velocity co about / with the simul- 

 taneous oblique linear velocity v is therefore equivalent to the 

 angular velocity co about V with the simultaneous linear velocity 

 V Q parallel to /'. 



295. When the rigid body has two simultaneous angular 

 velocities a> lt co 2 about skew axes l lt / 2 , the reduction is best made 

 by replacing co 2 about / 2 by an equal angular velocity o) 2 about a 

 parallel axis V intersecting t lt in combination with a linear 

 velocity v=pco <1 perpendicular to the plane of / 2 and /' (Art. 257). 

 The angular velocities co^ about / : and o> 2 about /' combine (by 

 Art. 292) into a singular angular velocity whose rotor is the 

 geometric sum of eoj and o> 2 . The case is therefore reduced to 

 the preceding one. 



296. It follows from the preceding articles that any number 

 of simultaneous linear and angular velocities can always be 

 combined into a single twist-velocity about the central axis. 





