252 ON A DYNAMICAL TOP. 



We shall call this axis of angular momentum the invariable axis. It is 

 perpendicular to what has been called the invariable plane. Poins6t calls it 

 the axis of the couple of impulsion. The direction-cosines of this axis in the 

 body are, 



, AQ>I B(o t 



= ' = ~> * 



Since I, m and n vary during the motion, we need some additional 

 condition to determine the relation between them. We find this in the property 

 of the vis viva of a system of invariable form in which there is no friction. 

 The vis viva of such a system must be constant. We express this in the 

 equation 



Atf + B^+Cu^V ................................. (2). 



Substituting the values of ,, o> 2 , to, in terms of I, m, n, 



Z* m| n'_JT 

 A + B + C~H>' 



Let ^=a\ B = ^' C =C> ' H t = e '' 



and this equation becomes 



a'P+fc'm'+c'n^e 1 ................................ (3), 



and the equation to the cone, described by the invariable axis within the 

 body, is 



(a 1 - e*) x> + (b t -e t ) tf + (c'-e')z s = ...................... (4). 



The intersections of this cone with planes perpendicular to the principal 

 axes are found by putting x, y, or z, constant in this equation. By giving 

 e various values, all the different paths of the pole of the invariable axis, 

 corresponding to different initial circumstances, may be traced. 



*In the figures, I have supposed a" = 100, 6 J =107, and c*=110. The 

 first figure represents a section of the various cones by a plane perpendicular 

 to the axis of x, which is that of greatest moment of inertia. These sections 

 are ellipses having their major axis parallel to the axis of 6. The value of e' 

 corresponding to each of these curves is indicated by figures beside the curve. 

 The ellipticity increases with the size of the ellipse, so that the section 

 corresponding to e*=107 would be two parallel straight lines (beyond the bounds 

 of the figure), after which the sections would be hyperbolas. 



