THE ORBITS OF THE EIGHT PRINCIPAL PLANETS. 115 



For the root g,= 2".916082, we get, 



,0.3678921 /r . _^ 54480 lv-. _580.9484 Ar/r2 



Ce 1 nlO 6 ' "fl -I mo ''6 > Z "'" " 



JQ1 



10 11 



Whence /? 6 =133 56' 10".8; and log. ^'"=6. 9441 833. 



N e =+0.003128, N 6 ir =+0.0008794, 



#' =+0.001811, N t r =+0.0007180, 

 JV 6 " =+0.001623, 

 JV 6 '"=+0.001156, 



= 0.0176872, 



2V m =+0.0019010. 



"or the root # 7 = 25".934567, we get, 



0.2996623 __.. , 0.2203054 AT , r 59.03157 



w* . /V i/ 1 Ar 5* 



"~JQ10 ^'7 > 111 H 1Q10 - tv ? ' ^ 1Q1 



. 



Xf l 



Whence ^=306 19' 21".2; log. ^/ r =7.7993771. 



N 7 =0.0002652, JV 7 / F =+0.00630053, 



JV 7 ' =0.0002932, N, r =0.0156928, 



JV 7 " =0.0027275, N 7 TI =+0.0006890, 



JV 7 '"= 0.0092499, JV 7 r/ '=+0.00007720. 



If these values be substituted in equations (F), we shall have the complete values 

 of q, q', q", &c., p, p', p", &c., from which we can obtain the inclination of the orbits 

 of all the planets to the fixed ecliptic of 1850, and the longitudes of the nodes, on 

 the same plane and referred to the equinox of 1850, by the formulae 



tan <2>=n 



tan 6=-^. 



(412) 



8. If we how substitute in equations (F), the values of q and p, we shall get 

 9 =tan $ cos 0=^cos (^+ / 3)+^ 1 cos fat+PJ+N, cos (g^+&)+ &c.; (413) 

 p=tan <p sin e=Nsw (gt+^+Ni sin (gj+fid+Nt sin (^+ft)+ &c - ( 414 ) 



Multiplying equations (413) by sin (#<+/3), and (414) by cos (#<+/3), we shall 

 get, by adding their products, and reducing 



tan (?) sin (0gt(3)=N l sin { (g l g)t-{-(3 l (3 \ +N 2 sin {(g z g)t ) ... 



If we multiply (413) by cos (gt-\-(3), and (414) by sin 

 adding their products, and reducing 

 tan q> cos (6 gt /3)= 



, we shall get, by 



cos 



cos z 



Dividing equation (415) by (416) we eliminate tan$, and find, 

 tan(0 gt /3)= 



sn - 



sn t - 



- &c. 



cos r 1 _ 



cos z - 



-3 &c 



:} <417) 



When the sum ^+^+^3+ &c. of the coefficients of the cosines of the deno- 

 minator, taken positively, is less than N, tan (6 gt /3) cannot become infinite ; 

 the angle (0gt(3) cannot become a right angle: consequently, the mean motion 



