2 PRECESSION OF THE EQUINOXES AND NUTATION 



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

 those of the moving axes. 



For tlie purpose of determining tlie positions of the axes Ox^, Oij,, and Ozj^, with 

 reference to tile (lixed in space) axes Ox, Oy, Oz, three auxiliary angles are used. 



If we suppose the moving plane of a-j y^, at the instant considered, to intersect 

 i\w fixed plane of xy in the line N N' and call the angle xON=^i', and the angle 

 b(>t\vccn tlie planes xy and x^y^ (or the angle zOz{)=Q, and the angle NOxj^=z^ (in 

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

 will determine the positions of the axes Ox^, Oyi, Oz^ (and hence of the body) at 

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

 about the ax(>s of a-j, v/i, and z^, may be expressed in terms of them and of their 

 differential coefhcients. 



A^'hen a body is a solid of revolution, revolving with an angular velocity «, 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 jjaper referred to)^ take the form 



sm^ d- --=— — (cos Q — cos w) 

 at A 



df = ndt -\-cos Od^^ 

 In which 



M is the mass of the body. 



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

 C " " " " its axis of figure. 



(J the force of gravity. 



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

 (.) 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 fio-ure. 



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



My '■lA^g~ X 



3. sin^ Q^^ = ^1 [sin^ 0-2/3^ (cos 0-cos ca)] (cos 0-cos c.). 



df ~ X 

 and the first equation (1) becomes 



sin^olj'^^ 2/3^1 (cos e-cosc) 



quantity /3=^ (?_ *^« , 



'■iA\ g~2y^W^' "^^y "^ ^'^^'y S'"^^* "^^ consequence of the 



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



4. 



The 



nowm""^^"' '''"."" ''''•^ """' °'°'''^' '''^''' ^'■°" P^'^^o" ^^ f" «« equations (of that paper) (9) 

 (10), (II), corrcspondrnff to n) (S) C9^ of this- >,„+ 4V,^ i , , ""' ^'^' """ I'''P*'n ('';. 



1 fa" '-'^' ^°''' v^-* o'this; but the subsequeutdevelopiucuts were original. 



