Rotation of a Solid Body round a Fixed Point. 335 



from the preceding equation by replacing C' and tan by their 

 values /iP 2 , and -= ; Q being the line EP. 



We thus obtain finally the centrifugal couple lying in the 

 plane XOZ, and expressed by the equation 



w*lxzdm = - ^PQ. (8) 



It thus appears that the centrifugal couple lies in the plane of 

 radius vector and perpendicular, is proportional to the area of 

 the triangle EOF, and has a direction opposite to the direction 

 of rotation. 



V. To FIND THE EELATION BETWEEN THE PLANE OF PRINCIPAL 

 MOMENTS AND THE Axis OF EOTATION AT ANY INSTANT. 



The motion of the body at any instant consists of a rotation 

 of a certain magnitude round a certain axis ; this rotation might 

 be produced by an impulsive couple of a determinate magnitude 

 and direction. The ^statical impulsive couple thus conceived is 

 the couple of principal moments. Let this couple be represented 

 by 6r, and act round the axis OE (fig. 1, p. 334) ; then the cor- 

 responding axis of rotation will be the perpendicular OP, and 

 the relation between 6r and w may be thus found : Let the axes 

 of co-ordinates be the axes of the ellipsoid (4), the radius vector 

 being determined by the angles (A, n, v), and the axes of rota- 

 tion by the angles (a, |3, 7). From mechanical considerations 

 we obtain the equations 



G cos X = Ap = juw a z cos a ; 

 G cos fj. = Bq = jua; b z cos/3 ; 

 G cos v = Cr = IULW c 2 cos 7. 

 Hence we obtain 



cos A _ a? cos a cos.^i _ b* cos/3 

 cos v c 2 cos 7' cos v c 2 cos 7* 



(9) 

 G 



