14G The Rev. Samuel Haughton's Account of 



this Hne and the original radius vector are axes of the section of the elhpsoid 

 made by their plane; at the point where the axis of the centrifugal couple pierces 

 the ellipsoid let a tangent plane be applied ; the perpendicular let fall on this 

 tangent plane is the axis of rotation produced by centrifugal forces. From the 

 construction it is evident that the plane of the second radius vector and per- 

 pendicular is perpendicular to the axis of G ; hence the axis of the centrifugal 

 couple and the axis of rotation produced by it, always lie in the plane of prin- 

 cipal moment. Two important corollaries follow from the theorem just de- 

 monstrated, in the case where no forces act: — First, the component of angular 

 velocity round the axis of primitive impulse is constant during the motion. 

 Secondly, the radius vector v/hich coincides with the axis of G is of constant 

 lenofth during the motion. The first theorem is obvious : for as the axis of 

 rotation produced by centrifugal force is always perpendicular to the axis of G, 

 it cannot alter the rotation round that axis. The second theorem follows from 

 equation (9), from which we deduce 



"- cos (/. = ^. (10) 



The left hand member of this equation is constant by the preceding theorem ; 

 and G is constant, since there is no external force ; therefore R is constant. 



As the axis of G is fixed in space, and the line B is constant, it is evident 

 that the axis of G will describe in the body the cone of the second degree, de- 

 termined by the intersection of the ellipsoid (4) with the sphere whose radius 

 is R. The equation of this cone is 



^^-^- + — 2v-r+-^^c =0. (11) 



As the axis of principal moments describes this cone in the body, it is accom- 

 panied by the axis of rotation, which is always the corresponding perpendicular 

 on tangent plane of the ellipsoid. The cone described by the axis of rotation 

 might be found thus. Let tangent planes be applied to the ellipsoid along the 

 spherical conic in which the cone (ll)cuts the ellipsoid. From the centre let fall 

 perpendiculars on these tangent planes ; the locus of these perpendiculars is 

 the required cone. 



