344 Rotation of a Solid Body round a Fixed Point. 



The position of the body in space is thus reduced to quadra- 

 tures ; but the problem may be solved more readily in the fol- 

 lowing manner. 



Second Method. 



The axis of principal moments, appearing to move in a 

 direction opposite to the rotation, describes in the body the cone 

 whose equation has been given (11). If the cone reciprocal to 

 this cone be described, one of its sides will lie in the fixed 

 plane, and the whole motion of the body in space will be the 

 same as the motion of this cone, which partly slides and partly 

 rolls on the fixed plane, the sliding motion being uniform. This 

 theorem is evident by resolving the angular velocity w into two 

 components, one round the axis of principal moments, and the 

 other in a direction perpendicular to this, round the side of the 

 reciprocal cone, which is in contact with the fixed plane. These 

 components are w cos and w sin ; w cos being constant and 

 producing the sliding motion, while <o sin <j> represents the an- 

 gular velocity round the side of the cone in contact with the 

 fixed plane. The angle described by the side of the reciprocal 

 cone in the fixed plane at the end of a given time is, therefore, 

 the algebraic sum of two angles, one of which is proportional to 

 the time, and the other is the angle described in the cone in con- 

 sequence of the rotation w sin 0, and is, therefore, measured by 

 the arc of a spherical conic. The position of the body at the 

 end of the time t is thus found : determine by equation (14) 

 the position of the axis of principal moments in the cone (11) ; 

 the corresponding position of the component axis of rotation in 

 the reciprocal cone is therefore known. Hence the angle de- 

 scribed in the time t in the fixed plane is 



(fjQ O 



- = 0* COS . t -r-. (19) 



The equation of the reciprocal cone is 



n 1 v % 7,2 7 ,2 ,.2-2 



y . _ _ ft (201 



+ + * ~ U< 



