Professor Mac Cullagh's Lectures on Rotation. 153 



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 th>s cone, which partly slides and partly rolls on tlie fixed plane the 

 shdmg motion being uniform. This theorem is evident by resolving the' an- 

 gukr velocity . into two components, one round the axis of principal moment<= 

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

 procal cone which is in contact with the fixed plane. These components are 

 u. cos and <« sm ; . cos being constant and producing the slidin.^ motion 

 while . sm represents the angular 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 an^le 

 described in the cone in consequence of the rotation a, sin 0, and is therefore 

 measured by the arc of a spherical conic. The position of the body at the end 

 of the time Ms 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 described in the time t in the fixed plane is 



G=|«.cos0fZ?±|^ = tt,cos0.«± ^. (19^ 



The equation of the reciprocal cone is 



E' - (r ^ Ji' -P+ W^rp = ^- (20) 



In (19) the positive or negative sign must be used according as E is le.s 

 or greater than the mean axis of the ellipsoid; this is evident from the com- 

 position of rotations, and from the consideration that in the former case the 

 axis of rotation falls inside the cone (11), while in the latter case it falls outside. 



^.— To find a Point in a given Axis of Rotation, which heing fized, the Axis iciU 



he permanerd. 



_ Let R'E" (fig. 4) be the given axis, round which tlie body revolves with a ro- 

 tation expressed by .■ describe the ellipsoid of gyration round the centre of 



VOL. XXII. J 



