343-] 



BODY WITH FIXED POINT. 



I8 9 



along the principal axes at O, The left-hand members reduce 

 also to a simple form if the differentiations indicated in (16) 

 be performed, the values for a lt # 2 , a 3 , b^ be substituted 

 from Art. 340, and the relations (9) be applied. As final result 

 we find Euler's equations : 



3 + (7 2 - /! 



3. CONTINUOUS MOTION WITHOUT FORCES. 



343. In the particular case when no external forces are act- 

 ing, the motion of a rigid body about a fixed point admits of an 

 elegant geometrical interpretation which is due to Poinsot. 



As there are no external forces, we have //"=o, and hence 

 H is constant in magnitude and direction. The plane of this 

 couple is the invariable plane (see Arts. 230-232) which always 

 exists in the case of no forces ; its vector indicates the invari- 

 able direction. 



The body can be replaced by its momental ellipsoid at O, and 

 the invariable plane can be imagined placed so as to be tan- 

 gent to this ellipsoid at a point P' 

 (Fig. 43). The radius vector OP' = p' 

 of the point of contact P' is the diam- 

 eter of the ellipsoid conjugate to the 

 invariable plane ; hence the line OP' 

 is the instantaneous axis / of the 

 rotation (Art. 321). 



Now it can be shown that the per- 

 pendicular distance q 1 of O from the 

 invariable plane (this plane being 

 always placed so as to be tangent 

 to the varying positions of the mo- 

 mental ellipsoid) is constant ; it then 

 follows at once that the motion of the body consists in the rolling 

 of its momental ellipsoid over the invariable plane. 



