4 PRECESSION OF THE EQUINOXES AND NUTATION 



If the value of u is not 90, the diameter of this circle will be 7 ^ sinw; but the 

 quantity - then measures an angle of an arc of a small circle having a radius 



=sin o; and the chord of the curve is reduced in the same proportion as its sagitta, 

 and the curve is still a cycloid. 



The axis of figure gyrates therefore about the vertical through Gas if it was attached 

 to the circumference of a small circle of the minute diameter specified, the centre 

 of which circles moves with a uniform horizontal velocity, which velocity is the 

 mean rate of gyration ; or the motion may be compared to that of a cone having its 



vertex at and diameter of base =7^ sin <j (the axis of figure constituting an 



element of this cone) rolling upon a fixed conical surface, all the elements of which 

 make, with the vertical, the angle ; but this imaginary cone %s not fixed in the body 

 (save in the exceptional case of the moments of inertia A and C being equal). For 



the rotary velocity of the body is n, while that of the cone is 2/2 [=__. 



*\/ A A. 



Since the rotary velocities of the body and cone are different, the instantaneous 

 axis cannot move along the chord of the cycloid, nor with uniform velocity. 



The common methods of investigating the Precession of the Equinoxes, founded 

 upon the incipient rate of motion of the instantaneous axis, involve this error, which 

 does not become apparent, simply because the moments of inertia A and C are, for 

 the earth, so nearly equal. 



The instantaneous axis will describe a prolate cycloid having the same chord as 



the common one (7,752)5 an d a sagitta = sin o (- l). 



The mean rate of gyration is given by the coefficient of t in equation (9); it is 

 12\*, or substituting values (2) for ft and A ; 



n. 



Cn 



Thus far I have supposed that at the origin of time, or at the moment when the 

 accelerating force commenced to act, the body had no other motion than a rotation, 

 n, about its axis of figure. It remains to prove, that if, at this instant, there are 

 small (compared to n) velocities about either or both the other principal axes, the 

 rate of gyration will be the same. 



The solution will be perfectly general if we suppose at this instant a velocity, m, 

 about the axis of x only, and assume at same moment 0=a, $=90, we should get, 

 instead of equations (3) and (4), the two following : 



19 siTv*fl M * T 2J % "A ICmn . CW, 



LJ. sin - s=i -^*- sin- 6 -- - sin ---- -- 2 - (cos cos to) 



m 2 (cos 0-j-cos u ) J (cos cos o) 

 sin 2 & -J- (cos 6 cos o)4-m sin o> 



