THE ORBITS OF THE EIGHT PRINCIPAL PLANETS. 115 



For the root g =— 2".916082, we get, 



, 0.3678921 0.354 4802 _580.9484 



io 10 ' io 10 ' — 10 



Whence &=133° 56' 10".8; and log. N S IT =6. 944 1833. 



JV a =+0.003128, Nf =+0.0008794, 



iV a ' =+0.001811, NJ =+0.0007180, 



NJ =+0.001623, N e rz =— 0.0176872, 



iV 6 '"= +0.00 1156, N/ 1 =+0.0019010. 



For the root </ 7 = — 25".934567, we get, 



0.2996623 Ar/r . 0.2203054 xr/r 59.03157 XT , r9 



^= ro ,0 ~ 7 ;2/7= ^ IO 55 — ^ 7 ; Zl= ~~ 1W~ N " • 



Whence /3 7 =306° 19' 21".2; log. iV/ r =7.7993771. 



N 7 =—0.0002652, iV/ r =+0.00630053, 



N,' =—0.0002932, N/ =—0.0156928, 



iV 7 " =— 0.0027275, N 7 " =+0.0006890, 



iV 7 '"=— 0.0092499, iV." '=+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 <P=Vy+?*; tan 0=p-^q. (412) 



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

 5=tan $ cos 0=iVcos ( flfi + / 3)+ j\r cos (^+ / #.)+ JV 2 C os (#,$+#,)+ &c; (413) 

 jp=tan $ sin 0=J\Tsin [gt+P)^-^ sin (Vi*+/?i)+^2 sin (gd-\-p 2 )-\- &c. (414) 



Multiplying equations (413) by sin (gt-\-(3), and (414) by — cos (gt-\-p), we shall 

 get, by adding their products, and reducing 



tan sin (0— ^— /?)=iV 1 sin j ^—gy+p.—p j +iV 2 sin {{g,—g)t 1 ... 



+&-/3j+&c.i 



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

 adding their products, and reducing 



tantf>cos(0— gt— p)=N-\-N l coa ]( 9l — g)t+ fa— ft \-\-N 2 cos \(g a —g)t\ (4l6) 



Dividing equation (415) by (416) we eliminate tan<?>, and find, 

 tan(0— gt—@)= 



N, sin | (fr-g y+fr-p I +JV, sin | (g 2 -g)t+p 2 ~p I + &c. 1 41 



N+fi COS \ (g.-gy+fa-p I +N 2 COS { (g 2 -g)t+p 2 -(j } + &c. J k 



When the sum iY r 1 +iV 2 +iV3+ &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 (0 — gt — p) cannot become a right angle: consequently, the mean motion 



