KINETICS OF A RIGID BODY. [323. 



323. If the principal axes at O be taken as axes of co-ordi- 

 nates, we have to write <o v <o 2 , o> 3 for <a x , <o y , ta z ; H lt /7 2 , 7/ 3 for 

 .H M H y , H n \ 7 lf 7 2 , 7 3 for A t B, C, while Z> = o, =o, F=o. 

 Thus the relations (5) give // 1 = 7 1 eo 1 , 7/ 2 = 7 2 o> 2 , 7/ 3 = 7 3 a> 3 , whence 



SL + SL- ( l6) 



and 7^= ^ (T^wj 4- 7 2 o) 2 + 7 3 o> 3 ) (17) 



(18) 



For the angle < between 7f and o>, we have 



COS ffl = -*-"* ^ ~. = ^ * rr 



7/0) 7/o> 



2. CONTINUOUS MOTION UNDER ANY FORCES. 



324. We now proceed to consider the motion of a rigid body 

 with a fixed point when acted upon by any forces. 



For the fixed point O as origin, the external forces reduce to 

 a resultant R and a couple H. While the force R is taken up 

 by the fixed point, the effect of the couple consists in changing 

 the angular velocity o> about the instantaneous axis /, which 

 exists at the time /, to the angular velocity o>4-</o) about 

 another instantaneous axis /', which determines the motion of 

 the body at the time t + dt. The point O being fixed, both 

 axes, / and /, pass through it ; and by Part I., Art. 303, the 

 acceleration of any point (x, y, z) of the body has the following 

 components parallel to rectangular axes fixed in the body and 

 moving with it : 



x = a> x (m x x 4- 



y = & y (co x x+c0 y y + co^)(o 2 y + Q) g xQ) x z, (l) 



'z = c0 g ((t) x x 4- Q) y y -h o)^) o) 2 ^ 4- co x y ufx. 



