328-] 



BODY WITH FIXED POINT. 



179 



w r ff y 



other has the components ^ y H z (a z f y , u z H x u> x H z , 



Each of these components can be inter- 



preted geometrically, if we imagine the vector 



H of the impulsive couple drawn from O as 



origin, so that the co-ordinates of its extremity 



are H x , H y , H z (Fig. 40). The time-deriva- 



tives of these co-ordinates are the velocities 



of the extremity of the vector H with respect 



to the axes of co-ordinates which, it will be 



remembered, are fixed in the body. Hence 



that component of H which is due to the 



angular acceleration is the relative velocity of 



the extremity of H with respect to the body. 



The other component, which is due to the centripetal acceleration, 

 evidently represents the linear velocity, arising from the angular velocity 

 <o, of the point of the body that coincides at the time / with the same 

 extremity of the vector H. 



It follows that the vector H represents in magnitude and direction 

 the absolute velocity of the extremity of the vector H ' in other words, 

 H is geometrically equal to the geometrical increment of H divided by 

 the element of time. This was to be expected, and might indeed be 

 taken as starting-point for deriving the equations (3). 



Fig. 49. 



328. Let us now select as axes of co-ordinates the principal 

 axes at O. According to our usual notation, we have then to 

 exchange the subscripts x, y, z for i, 2, 3. Moreover, as shown 

 in Art. 319, H^I^ ff 2 = I 2 co 2 , /7 3 = 7 3 a> 3 , where 7 lf 7 2 , 7 3 are 

 the principal moments of inertia at O. Thus the equations (3) 

 reduce to the following : 



a) l + (7 8 - 



(4) 



7 3 d> 3 + (7 2 - 7^ o) 1 c 2 = 7/g. 



These are Euler's equations of motion. Their solution gives 

 o) 2 , o> 3 as functions of the time t. 



