!88 KINETICS OF A RIGID BODY. [341. 



341. From (7) and (15) we now find, if we remember that 

 2my'z' = o, 2mz'x' = o, ^m^y = o t since Ox\ Oy\ Oz' are prin- 

 cipal axes at O : 



2m (y'z zy)=- (b 2 c z 2 



or applying the relations (13) and denoting the principal 

 moments of inertia at O by I v 7 2 , / 3 : 



*%m(yz zy) = a l l l co 1 -f- # 2 / 2 ft> 2 + #3/30)3. 



The quantities *2m(zxX!5) and ^m(xyyx) are obtained from 

 this result by cyclical permutation of the letters a, b> c. 

 Thus the equations (7), Art. 224, assume the form : 



-rX^i/i! + a 2 f 2 a) 2 + tf 3 / 3 &> 3 ) = H* 

 at 



^' ( ! 6 > 



-- (^x/!! 4- ^22 + ^B^B) = H * 

 at 







The geometrical meaning of these equations is apparent. 

 /i!, / 2 w 2 , / 3 w 3 are the components along the principal axes of 

 the vector of the resultant impulsive couple H (Art. 319); 

 hence <2 1 / 1 ft) 1 -f-^ 2 / 2 ft) 2 4-^ 3 / 3 a) 3 is the component of H along the 

 fixed axis Ox\ the equations (16) express therefore the fact 

 that H is geometrically equal to the geometrical derivative 

 of H with respect to the time (see Art. 327) ; they can be 

 written in the form 



dH z TT 



342. If the equations (16) be multiplied first by a l9 d v c lt 

 then by a 2 , 2 , <: 2 , finally by ^ 3 , 3 , c 3 , and each time added, the 

 right-hand members of the resulting equations will evidently 

 represent ff v ' H v ff s , respectively, i.e. the components of H 



