562 PKOFESSOB, MAXWELL ON A DYNAMICAL TOP. 



mum. These two are at right angles, and the third axis is perpendicular to 

 their plan, and is called the mean axis. 



* Let A, B, C be the moments of inertia about the principal axis through the 

 centre of gravity, taken in order of magnitude, and let Wj w 2 o> 3 be the angular 

 velocities about them, then the angular momentum will be Aw l5 Bw 2 and C&> 3 . 



Angular momentum may be compounded like forces or velocities, by the law of 

 the " parallelogram," and since these three are at right angles to each other, their 

 resultant is 



\ZA2w, 2 + B 2 w 2 2 + C 2 w 3 2 =H .... (1) 



and this must be constant, both in magnitude and direction in space, since no 

 external forces act on the body. 



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

 pendicular to what has been called the invariable plane. Poinsot calls it the axis 

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



7 Aw. Bw„ Cw„ 



l =~W W= TT n= TT 



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 



Aw 1 2 + Bw 2 2 + Cw 3 2 = V; ... (2) 



Substituting the values of w 1 w 2 w 3 in terms of /, m, n 



I 2 m 2 n 2 _ V 

 A + B" + "C~H 2 



Let JL_ a s !_-7>2 L- C 2 V- e 2 

 A' B ' C ' H 2 



and this equation becomes 



a 2 l 2 + b 2 m, 2 + c 2 n 2 =e 7 (3) 



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



is 



(a 2 -e' 2 y+(b 2 -e 2 )y 2 + {c 2 -e 2 )z 2 = ... (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 8 = 100, & 2 =107, and c 2 = 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 b. The value of tf corresponding 

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



