THE ORBITS OF THE EIGHT PRINCIPAL PLANETS. 



13 



4 1 7jA-'-|-6 1 =0, 



3^ w -fi a =o, 



If we now suppose b, b, b", &c. to be equal to nothing, and eliminate IV, N", and 

 N" from equations (B"), and N\ N", and JV r// from equations (B'"), the resulting 

 equations will be divisible by N and N' v respectively ; and we shall have 



+[iTT][TT7][T]nii^+ir^r^r^ 



, -2 2 , ii o , 3 



3,2 2,0 0,1 1,3 3,0 0,1 1,2 2,3 + 3, 2 2,1 1,0 0,3 



4,4 .I,* 8,6 7,7 7,6 6,7 4,4 5,5 7,8 5,7 4,4 8,6 7,4 4,7 a, 5 6, 



6,05,64,47,7 6,44,65,57,7 .',,44,56,67,7 







7,b6,44,S-,,7 7,4 4.5 -,,66,7 -4-7,6 6,5 5,4 4,7 



Each of these equations is evidently of the fourth degree in g, and consequently 

 has four roots, which may be found by any of the ordinary methods of finding 

 the roots of numerical equations. These roots will be only approximate, because 

 we have neglected b, b\ b", &c., in the determination of equations (29) and (30). 

 If we substitute the approximate roots derived from equation (29) in any three of 

 equations (5"), we can find by elimination the values of N\ N", and N'" in terms of 

 N, which remains indeterminate. When N', N", and N'" have been thus deter- 

 mined we must substitute their values in equations (28), and we shall obtain the 

 values J 1? 5 25 ^3, and b^ in terms of N; and these quantities are then to be substituted 

 in equations (-B'"), together with the corresponding value of g; and we shall then 

 obtain the values of N'\ N r , N VI , and N T ", in terms N. But instead of performing 

 this operation separately for each of the roots, in the manner described above, it is 

 better to deduce a system of algebraic equations, not only for the purpose of facili- 

 tating the numerical calculations, but also for the purpose of devising checks to 

 the accuracy of the different parts of the computations. 



10. If we now assume the following quantities 



(32) 



