THE ORBITS OF THE EIGHT PRINCIPAL PLANETS. 11 



8. The values of (0,1), (0,2), &c., [oTT], [oTa], &c., (1,0), (1,2), &c., QTo], [T^\, &c., 

 being substituted in equations (B), supposing ^, ^', ^", &c., to be equal to nothing, 

 we shall have a series of numerical equations which are perfectly symmetrical in 

 form, between g and the unknown quantities N, N', N", &c. If we then eliminate 

 N, N', N" from these equations, we shall obtain a final equation in g, of a degree 

 equal to the number of original equations. The construction of this final equation 

 in g is the most delicate and intricate problem connected with the actual determi- 

 nation of the secular inequalities. Theoretically speaking, this equation may be 

 formed by eliminating the quantities N, N', N", &c., by the ordinary methods of 

 elimination in algebra. But this method, though direct and simple in theory, leads 

 to impracticable operations when we attempt to apply it to the formation of the 

 equation of the eighth degree which is necessary in the simultaneous determination 

 of the secular inequalities of the eight principal planets. The only actual merit 

 of this method seems to be that of leaving the algebraic values of N',N", N'", &c., 

 in the successive eliminations, in very good form for computation, when the value 

 of g has been determined ; while its defects are twofold, as follows : First, it intro- 

 duces foreign facts depending on g, which raise the final equation in g, to a very 

 high degree ; and secondly, it necessitates the employment of a very great number 

 of decimals in the successive eliminations, in order to obtain a near approximation 

 to the correct value of the final equation. The method of determinants enables us 

 to form the final equation in g without actually performing the eliminations of the 

 unknown quantities N, N', N", &c. It also enables us to estimate, in advance, the 

 exact amount of labor necessary for forming the final equation arising from any 

 number of linear symmetrical equations. In the year 1860, we published a short 

 paper on this interesting branch of analysis in GOULD'S Astronomical Journal, Vol. 

 VI. From the explanation and formulae there given, it follows that each of the 

 given equations contains one binomial term of the form g a, and that each term 

 of the final equation contains one factor from each of the given equations ; and 

 also that the whole number of terms in the final equation is equal to the continued 

 product of the numbers 1, 2, 3, 4, &c., to n inclusive; n denoting the number of 

 given equations. In the present case n is equal to eight ; there being one equation 

 corresponding to each of the eight principal planets. The whole number of terms 

 in the equation of the eighth degree is therefore equal to 1.2.3.4.5.6.7.8 = 

 40320. There are therefore 40320 distinct terms in the equation of the eighth 

 degree, each of which contains eight factors which are either monomial or binomial. 

 They are distributed in the following order : 



1 term having 8 binomial factors producing 9 monomial terms. 

 28 terms " 6 " " " 196 



112 " " 5 " " " 672 " " 



630 " " 4 ". " " 3150 " " 



2464 " " 3 " " " 9856 " " 



7420 " " 2 " " " 22260 " 



14832 " 1 " " " 29664 



14833 " without binomial factors " 14833 



Total 40320 80640 



