304 ON THE GENERAL VALUES OF THE OBLIQUITY OF THE ECLIPTIC. [41 



ecliptic at any time, the rate of the luni-solar precession at that time 

 during a Julian year will be represented by ccosco, where c = 54"-94625 

 nearly. 



Now let ON'N be the fixed plane of reference, which may be either 

 the ecliptic at a given epoch, or, better still, the invariable plane of the 

 system, or any other arbitrary fixed plane. 



Also let N'E be the position of the ecliptic 1 



, , 7ll ,, c ,, \ at any time t, 



and NE that ot the equator 



so that the point E is the autumnal equinox at that time. ON=<f>, 

 ON' = <$>', O being a fixed point, 6 and 6' the inclination of the equator 

 and ecliptic respectively to the fixed plane, and co the angle N'EN, or 

 the obliquity of the ecliptic at time t. Also let NE = \. Then the 

 quantities p = tan 6' sin c// and q tan 0' cos c// are known in terms of / 

 from the theory of the secular variations of the plane of the Earth's 

 orbit, and & may be considered as a small quantity of the first order, 

 the square of which we propose to take into account. 



In the triangle N'EN we have 



cos co = cos 6 cos 6 1 + sin 6 sin 6' cos (c/> c//), 

 sin co cos X = sin 9 cos & cos 6 sin 9' cos (c/> <'), 

 sin co sin X = sin 6' sin^(c/> c//), 

 which give co and X when 9 and c/> are known. 



From the instantaneous motion of the equator with reference to the 

 ecliptic at time t, supposed for an instant to be fixed, it is easily seen 

 that we have 



r- = c . cos co sin co cos X, 

 at sm 6 



dO 



= c cos co sin co sin X, 



