KINEMATICS. [290. 



direction cosines of the principal normal. The formulae show 

 therefore that the acceleration j consists of two components, 



^ !=^ along the tangent, and L/!^Y.!? along the normal. 

 dt* dt p\dt J p 



2. VELOCITIES IN THE RIGID BODY. 



290. When the motion of a rigid body is a translation, all 

 points of the body have at any instant equal and parallel veloc- 

 ities (Art. 250). The velocity v = ds/dt of any one point can 

 therefore be called the velocity of the body. The body can be 

 subjected at a given instant to several velocities of translation, 

 and the resultant velocity is found by the geometrical addition 

 of the vectors representing the component velocities. 



291. When a rigid body rotates at the time t about an instan- 

 taneous axis /, all its points (excepting those on the axis) 

 describe infinitesimal arcs of circles of angle dO, and the 

 angular velocity a> = dd/dt of any point of the body may be 

 called the angular velocity of the body. This angular velocity 

 can be represented geometrically by its rotor o> laid off on the 

 axis /(Arts. 68, 69, 252). 



As this rotor is proportional to the infinitesimal angle o 

 rotation dd, the propositions proved in Arts. 62, 66, 67, 68, fo 

 the composition and resolution of infinitesimal rotations can b 

 applied directly to angular velocities. The propositions refer 

 ring to parallel axes have been discussed in Arts. 254-257. 



292. If in Art. 62 we divide equation (i') by dt 2 and divide 

 the denominators of equation (2') by dt, we obtain 



ft, 2 = , 1 2 + ft, 2 2 + 2ft) 1 ft) 2 cos (/^ (i 



sin (/!/) _ sin (// 2 ) _ sin (// 2 ) , 



ft) 2 ft)} ft> 



The meaning of these equations can be stated as follows. Le 

 a rigid body be subjected simultaneously to two angula 



