

2 9 3-] THE RIGID BODY. ^5 



velocities about intersecting axes, ^ about / x and <o 2 about / 2 . 

 Represent these angular velocities by their rotors co v &> 2 laid 

 off on the axes / p / 2 from their point of intersection O and 

 construct their geometric sum w ; that is, form the diagonal 

 of the parallelogram whose adjacent sides are co ly co 2 . Then co 

 is the rotor of the resulting angular velocity. 



This proposition is known as the parallelogram of angular 

 velocities. 



It follows that the resultant of any number of simultaneous 

 angular velocities whose axes all intersect in the same point is 

 a single angular velocity whose rotor is found by geometrically 

 adding the rotors of the components. 



293. Conversely, an angular velocity co about an axis / can 

 always be replaced, in an infinite number of ways, by two (or 

 more) angular velocities whose geometric sum is o>, about two 

 (or more) axes passing through any point O of / and lying in 

 the same plane with /. 



Thus, for instance, the angular velocity co about the instan- 

 taneous axis / can be resolved into three components co x , co y , w z 

 about three rectangular axes Ox y Oy, Oz passing through any 

 point O of /, and we have 



a> 2 = < z 2 + aV J + aV J . (3) 



The linear velocity v of any point P of a body rotating with 

 angular velocity w about the axis / can be expressed by means 

 of the components &> x , co y , w z of co and the co-ordinates x t y, z of 

 the point P. The component co x produces at P a velocity whose 

 components along the axes Ox, Oy, Oz are o, co^, w x y\ simi- 

 larly, o)y gives the components cOyZ, o, co y x; and co z gives w z y t 

 a> 2 x, o. Hence, combining the terms that lie along the same 

 axis, the components of the velocity v of the point P are 



(4) 



