

2 PRECESSION OF THE EQUINOXES AND NUTATION 



body's motion. The position of the body at any instant of time is determined by 

 those of ihe moving axes. 



For the .purpose of determining the positions of the axes Ox, Oy^ and Oz l5 with 

 ;Tgefereiioe-te 'the (fixed in space) axes Ox, Oy, Oz, three auxiliary angles are used. 



If we suppose the moving plane of x l y v at the instant considered, to intersect 

 the fixed plane of xy in the line NN' and call the angle xON=-^, and the angle 

 between the planes xy and x l y l (or the angle zOz^^O, and the angle N0x=q> (in 

 the figure these angles are supposed acute at the instant taken), these three angles 

 will determine the positions of the axes Ox^ Oy^ Oz 1 (and hence of the body) at 

 any instant, and will themselves be functions of the time . and the rotary velocities 

 about the axes of x y lt and z l5 may be expressed in terms of them and of their 

 differential coefficients. 



When a body is a solid of revolution, revolving with an angular velocity n, about 

 its axis of figure, and acted upon by the accelerating force of gravity (the fixed 

 point being in the axis of figure), the general equations of rotary motion (by 

 processes fully developed in the paper referred to) 1 take the form 



sin 2 6^=^ (cos 0-cos o>) 



d<> = ndt-{-co$ 

 In which 



M is the mass of the body. 



A its moment of inertia about an equatorial axis through 0. 

 G " " " " its axis of figure. 



g the force of gravity. 



y the distance OG from centre of gravity to the point of support, 

 o the initial value of 0, or its value at the instant when the body has no other 

 motion than the rotation n about its axis of figure. 



Eliminating -^ between the first two equations (1), and putting 



0/32 



O " i ~J '-' '* *>IJ 



2. ^ and =, we get 



3. sin 2 0- = [sin 2 2/3 2 (cos cos o)] (cos 6 cos a), 



Ctrl /+ 



and the first equation (1) becomes 



4. sin 2 0^=2/3 6 (cos 0-cos a)) 



dt \^ 



Fhe quantity ff= L ^ .^^, ma y be very great in consequence of the 

 rotary velocity, n, being great, or (n being small) in consequence of the ratio 



1 The analysis therein used was mostly taken from Poisson as far as equations (of that paper) (9), 

 (10), (11), corresponding to (7), (8), (9) of this; but the subsequent developments were original. 



