THE ORBITS OF THE EIGHT PRINCIPAL PLANETS. 



15 



(46) 





1 tun ^jes 3 ' 



3 1 E3 m i[i3 4 ' 



2 1 [EH u ~ ' J E3 



K48) 



f"=B+E"+F'; 



fi =B 2 -\-E i -] r F 2 ; 



/' r =S + J B'"+ J F"; 



/"=£"+ (7" +£'; 

 / r =5"+C""+^'"; 



fz=Bz-\-C z -\-E 2 \ 

 f,=B z -\-C,+F,; 



(49) 



(50) 



(51) 



(52) 



we shall have the following system of equations for the rigorous determination of 

 <7, N', N", JV'", &c. — observing that the "terms of equations (29) and (30), which are 

 inclosed by braces, are reduced to single terms by means of equations (14) and (15). 



E3IIH1I1H1E3- miEg[oT3[IH]— [I3G3CE3EI!] 



- EgEgiiaETi]- ^[t^^^- ^[^[n3[i3 



- [^[^[in]C!n]+2[!3E3[L3C^+2[IZ]G^[]Z][IID 



+ E3E3CI3EI1+ EaEEIlEaEa— »EIlEIlIEIlEia 

 -aE3iiaEgi^-fl(^|^EiiEa 



EairgEgEg- GSEUEgEg^ E3E3E3EI] 



- E3ehei]eii- EgngiigEg- ehes^eii 



- E3E3E3[I3+2[i3EZ]EIi][III]+20[iI3EI][EI] l 



+2iiaii3[±ai^+2[ia[iaii3[z3+ E3E3E3[H3 



+ [E3EIJE3EI1+ [I3E3E3E3-2LL3E3E3EZ] 

 -2E3EI]iiZlEa-2E3EI]E!]E!l 



=(z»zi>a:s'^8);(53) 



: (Z4.^6,^Z7);( 54 ) 



