332.] BODY WITH FIXED POINT. 181 



the reduction of momenta in Art. 317. For, evidently, the vector 

 representing the centrifugal force muPr can be obtained by multiply- 

 ing the momentum mwr of the particle m by cu and turning it through 

 an angle of 90 in a sense opposite to that of the rotation <o. The 

 reduction to O gives therefore a resultant force M^r, in the ary-plane, 

 directed toward the projection of the centroid on this plane. The 

 resulting couple has its 2-component equal to zero since all the centrifu- 

 gal forces intersect the axis of z; the vector of the resulting couple 

 lies therefore in the xy- plane, has the magnitude &H xyy and is perpen- 

 dicular to the H xy in Fig. 37. 



331. The resultant vanishes only if ? = o, i.e. if the centroid lies on 

 the axis /; the couple vanishes if uH xy = <o 2 VZ> 2 -f- 2 = o, i.e. if the 

 axis / is a principal axis at O. It follows that the centrifugal forces 

 reduce to zero only if the axis of rotation is a principal centroidal axis ; 

 in this case the direction of the axis remains unchanged. 



By Art. 318 (see Fig. 37) we have H xy = 7/sin < ; hence the result- 

 ing couple of the centrifugal forces = w/f sin <, that is, its magnitude is 

 represented by the area of the parallelogram formed by the vectors H 

 and a) ; the vector of this couple as shown above is perpendicular to 

 this area. Projecting this parallelogram on any three rectangular co-ordi- 

 nate planes, with O as origin, we find, since <a x , <a y , <a z are the co-ordi- 

 nates of the extremity of the rotor <o, H M H y , H y those of the extremity 

 of the vector H drawn from O : 



<a e H y w y H Mt w x H g <o z H xt o> y H x <ajf r 

 This agrees with the results found in Arts. 326, 327. 



332. If the principal axes at O be taken as axes of co-ordinates, the 

 components of the resultant couple of the centrifugal forces become 



or, since ^i = /!(!>!, H^I^^ .// 3 =/ 3 eo 3 , 



(7 2 7 3 ) (0 2 0) 3 , (7 3 /i) ftfcCDi, (/i 



As the planes of these couples are perpendicular to the principal axes at 

 O, they produce during the element of time infinitesimal rotations about 

 these axes, whose angles are, by Art. 318 : 



/I 



