VELOCITIES, &c. WITH RESPECT TO AXES MOVEABLE IN SPACE. 



15 



ration due to the forces (/r) : lastly, if OF be perpendicular to the plane HOI, it is the axis 

 of acceleration of angular momentum in the moving body, and OL, the radius for which the 

 normal is parallel to OH, is the axis of angular acceleration due to the motion of the body (\). 

 Also we have the three equations for w, k, X, 



lit) = h cos HI, 



Kk = G cos GK, 



L\ = / cos FL, 

 where y= hw sin HI, 

 I, K, L denoting the moments of inertia about 01, OK, OL respectively. It will be observed 

 that 01 is the direction, to which the plane through O perpendicular to OH is diametral, and 

 that OL is the direction to which the plane HOI is diametral, hence OL lies in the plane 

 perpendicular to OH. Also if the rectangular planes HOI, FOL intersect in OM, it will 

 be seen that the axes* 01, OL, OM axe conjugate diameters of the central ellipsoid. 



30. We will develop the solution in the simpler case of OG coinciding with OH and 

 therefore OK with OL In this case OH remains fixed in space, and the motion of 01 is 

 conveniently referred to its motion in the plane HOI and the motion of that plane about OH. 



AT 



Let the conjugate radii 01, OL, OMhe denoted by r, r, r", then the moments of inertia 



about them are -n -f^ , -rr^ > by the property of the central ellipsoid : also let the angles HOI, 



FOL be denoted by 9, 6' : then our last equations become 



(1) a) = Ar*cos0, (2) k = Gr^ cos ^, (S) \ = (Aw sin 0) . r'^ cos ^. 

 Resolve co, k, \ along the axes OH, OM, OF; the component velocities are then w cos 6 

 along OH, w sin 9 along OM, and zero along OF, while the component accelerations are 

 K cos 9 along OH, Ksm 9 +X sin 9' along OM, and X cos 9^ along OF ; whence, by applying 

 either the equation (C) or the equations (£), 



d 



dt 



(w cos 9) =KCos9 = Gr^ cos'' 9, . 



.(*) 



* Hence if no forces act, the instantaneous motion of the axis of rotation OJ will be towards OL, the radius with 

 respect to which the plane HOI is diametral. ' 



