Mr Bennett, The Rotation of the Non-Spinning Gyrostat 71 



by any conical movement, alters the angle between the plane ah 

 and any diameter of the wheel by an angle equal to the solid angle 

 enclosed by the cone formed by the conical surface ab together 

 with the plane ha. 



§ 3. A geometrical integration of Euler's equation leads to the 

 same result as § 1. The axis, with its direction given by spherical 

 polar coordinates d and (/> (radial and azimuthal), generates a solid 

 angle 



(7 = J(1 -COS^).f/(/.. (1) 



The equation of motion, being 



(^ COS ^ + i^ = 0, (2) 



with zero initial values for ^ and ^, has as its integral 



<f> + i/j = or. (3) 



If the axis of reference ^ = is supposed (conveniently) external 

 to the cone then </> is zero finally as well as initially, and i/r is the 

 angle of resultant displacement of the wheel and is equal to the 

 solid angle a. 



If, more generally, the gyrostat has a constant spin Q. about its 

 axis, the Euler equation becomes 



cf) cos 9 + i[i = Q (4) 



with 4> + ip = a + Qt (5) 



as its integral. And the final rotation of the gyrostat is then given 

 by the solid angle of the cone described by the axis plus the time- 

 integral of the spin. It may be noticed that the angle (f) + i/j, with 

 a value independent of the choice of coordinates, gives in itself a 

 natural measure of the total rotation of the wheel, as followed and 

 estimated by projection on the plane 6 = tt/'I. For on that plane 

 the circular disc shows as an ellipse, with ^ as the azimuth of the 

 direction of the minor axis, and ^ as the eccentric angle, measured 

 from the minor axis, of the projection of the revolving diameter of 

 the wheel. A distant observer on the axis ^ = 0, able to distinguish 

 the two faces of the wheel, would in this way precisely reckon the 

 amount of rotation, whole turns and fractional. He does not give 

 merely the ultimate position, by naming a plane angle to a modulus 

 of four right angles, but assigns the multiple of the modulus neces- 

 sary for a correct account of the movement intervening between 

 the initial and final positions. 



A kinematic representation of the angle </> + «/' may be obtained 

 by supposing the circular rim of the disc to have rolling contact 

 with the rim of another equal disc whose plane keeps parallel to 

 the plane 6 = 7r/2. The angle of rotation of this latter disc about 

 its axis (which keeps the invariable direction ^ = 0) is then </> + ip. 



