[88 MOTION OF INVARIABLE AXIS. 267 



Similarly for the rotation around the T and Z-axes, 





dv^ 3*0-1 

 dt ~ T sin 2 (He) 



Looking at the signs of the numerators, we see that the invariable 

 axis rotates around the axis of greatest moment of inertia in the 

 direction of rotation, about the least axis in the direction opposite 

 to that of rotation, and about the mean axis according to the 

 value of d. 



If we mark off on the invariable axis a line of unit length, its 

 end describes a sphero-conic, the intersection of the invariable cone 



50) 

 with the sphere 



whose projections on planes perpendicular to the X and Z-axes are 

 ellipses, and perpendicular to the Y"-axis an hyperbola. The radius 

 vector of the X- projection is r x = sm(Hx) and since it turns with 



the angular velocity ^- it describes area at the rate 



55 ) 2 %%-*(^-i). 



The time of one revolution of the body turning with the velocity p 

 would be, if p were constant, t = 



The equation of the ellipse is obtained by eliminating x from 

 the equations of the sphero-conic as 



whose axes are 



and whose area is 

 57) 



