I2 6 KINETICS OF A RIGID BODY. [230. 



Now, if at any time t the centroid were taken as origin, so that 

 3/ = o, i=o, this equation would reduce to the form 



?,m(rt-tt)=jr f 



which is entirely independent of the co-ordinates of the cen- 

 troid. On the other hand, wherever the origin is taken, if 

 the centroid were a fixed point, the same equation would 

 be obtained. 



Similar considerations apply of course to the other two 

 equations (7). It follows that the motion of a rigid body relative- 

 to the centroid is the same as if the centroid were fixed. 



' 230. If, in particular, the resultant couple H be zero for any 

 particular origin O (which will be the case not only when all 

 external forces are zero, but also when the directions of all 

 forces pass through the point O), the equations (7) can be 

 integrated and give 



yx} C^ (10) 



where C v C v C s are constants of integration (comp. Art. 94) 

 Hence, if the external forces pass through a fixed point, the 

 angular momentum of the body about any line through this 

 point is constant ; if there are no external forces, the angular 

 momentum is constant for any line whatever. This is the 

 principle of the conservation of angular momentum. 



231. Another interpretation can be given to these equations 

 As shown in Arts. 88 and 91, the quantities y'z zy, zxxz 

 xyyx can be regarded as sectorial velocities. Thus, if the 

 radius vector, drawn from the origin to the particle m, be pro 

 jected on thejs'-plane, y'z zy is twice the sectorial velocity of this 

 radius vector in the jj/^-plane, \(ydz zdy) being the elementary 

 sector described in the element of time dt. Let us denote by 

 dS x the sum of all these elementary sectors for the various 

 particles, each multiplied by the mass of the particle ; and 



