ON A DYNAMICAL TOP. 255 



The motion is in the same direction as that of rotation, or in the opposite 

 direction, according as the axis of x is that of greatest or of least moment 

 of inertia. 



*Both in this case, and in that in which the three axes are unequal, the 

 motion of the invariable axis in the body may be rendered very slow by 

 diminishing the difference of the moments of inertia. The angular velocity of 

 the axis of x about the invariable axis in space is 



'a a (l-Z')' 



which is greater or less than o,, as e* is greater or less than a?, and, when 

 these quantities are nearly equal, is very nearly the same as ^ itself This 

 quantity indicates the rate of revolution of the axle of the top about its 

 mean position, and is very easily observed. 



*The instantaneous axis is not so easily observed. It revolves round the 

 invariable axis in the same time with the axis of x, at a distance which is very 

 small in the case when a, b, c, are nearly equal. From its rapid angular motion 

 in space, and its near coincidence with the invariable axis, there is no advantage 

 in studying its motion in the top. 



*By making the moments of inertia very unequal, and in definite proportion 

 to each other, and by drawing a few strong lines as diameters of the disc, the 

 combination of motions will produce an appearance of epicycloids, which are the 

 result of the continued intersection of the successive positions of these lines, and 

 the cusps of the epicycloids lie in the curve in which the instantaneous axis 

 travels. Some of the figures produced in this way are very pleasing. 



In order to illustrate the theory of rotation experimentally, we must have 

 a body balanced on its centre of gravity, and capable of having its principal 

 axes and moments of inertia altered in form and position within certain limits. 

 We must be able to make the axle of the instrument the greatest, least, or 

 mean principal axis, or to make it not a principal axis at all, and we must be 

 able to see the position of the invariable axis of rotation at any time. There 

 must be three adjustments to regulate the position of the centre of gravity, 

 three for the magnitudes of the moments of inertia, and three for the directions 

 of the principal axes, nine independent adjustments, which may be distributed 

 as we please among the screws of the instrument. 



