Rotation of a Solid Body round a Fixed Point. 337 



cannot alter the rotation round that axis. The second theorem 

 follows from equation (9), from which we deduce 



w cos = . (10) 



The left-hand member of this equation is constant by 'the pre- 

 ceding 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 R is constant, 

 it is evident that the axis of G will describe in the body the cone 

 of the second degree, determined by the intersection of the ellip- 

 soid (4) with the sphere whose radius is R. The equation of 

 this cone is 



R*-a* , R*-V . R*-c* , 

 - - x* + - y 2 + - 2 2 = 0, (11) 



2 z * 



As the axis of principal moments describes this cone in the body, 

 it is accompanied 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 sphe- 

 rical conic in which the cone (11) cuts the ellipsoid. From the 

 centre let fall perpendiculars on these tangent planes ; the locus 

 of these perpendiculars is the required cone. 



VII. THE Axis OF PRINCIPAL MOMENTS is FIXED IN 



SPACE. 



This is evident from D'Alembert's principle, but may be 

 shown by geometrical considerations in the particular case under 

 consideration. The axis varies in position in the body, in con- 

 sequence of the centrifugal couple, which must be compounded 

 with the impressed couple at each instant. Eeferring to equa- 

 tion (8), the value of the centrifugal couple is - fivfPQdt, the 

 principal moment being G = pwPjR, (vid. (9)). Hence the angle 



