THE ORBITS OF THE EIGHT PRINCIPAL PLANETS. 169 

 Therefore we shall have 



'= A" sn 



'= JV' cos 



sin 

 :) sin 

 sin 

 cos 

 cos 

 -\-N 6 cos (s^+ft ft 



sin 



sin 

 sin 

 sin 

 cos 

 cos 

 cos 



sin 

 cos 



ft); 



(545) 



(546) 



Substituting for p and q their values given by the first members of equations 



(537), we shall easily find 



sin (j!> sin (f> gl {3)=^ sin J (g l g)t-\~Pi P \ -\-N% sin j (g 2 g)t-\-@2 /? ! 'J 



-f-JVgsin j((/ 3 g)t-\-(3 3 ^l-l-JVjsin \(g 6 g)t-\-p t (3\ > (547) 

 -\-N 6 sin { (g 6 g)t-\-fis P \ -\-Ni sin \ (g 7 g)t-\-(3 7 (3 \ ) 



sin eft, cos (0 gt /3) 1 



$\ \ (548> 



COS (g l - 

 COS 1 (g a - 

 COS t - 



COS (gr.- 

 COS | (y._ 

 COS - 



From these equations it is easy to show that the mean motion of is equal to 

 gt when N exceeds the sum of the coefficients of the cosines, all taken positively. 

 We shall also have 



maxmum <po= 

 and minimum. =N 



T; ) 

 ^. j 



4. If we now substitute in these equations the values given in Chapter II, 7, 

 we shall obtain the following maxima, minima, and mean motions. 



22 March, 1872. 



