THE ORBITS OF THE EIGHT PRINCIPAL TLANETS. 



33 



But each of the factors of /, I', I', &c, in the terms of the second member of this 

 equation, which are independent of <j, becomes nothing by means of equations (9) ; 

 and we shall therefore have 



i) /i /It ' m ft fir ' 



na dt 



n'a' dt 



n"a" dt 



.„ m . Tn , in in , --.,„„„ in . _ 



— a < M \- Nl — ; , + NT „- „ 4- NT- ,„ ,„+ &c. 



na na na n a ' 



(132) 



If we substitute in this equation the preceding values of h, h', 7b", Sec, 1, J', I", Sec , 

 we shall have, by comparing the coefficients of the same cosines, 



jyjVi m --+NN;™ ,+N"N{' n ^,,+N m N? ™ „,+ &c.^0; 

 na n a n a n a 



NN 2 —- +N'NJ ni r-,-\-N'N:",-„+N''Nr '"'",„+ &c.=0 ; 

 " na ' na " n a ' n a 



NN.—A-N'N^ m ' ,+N"N!' f n -\-N'"N^"-^ m 4- &c.=0 ; 

 na na n a n a 



&c. 



(133) 



If Ave now multiply the preceding values of 7b, h', 7b", Sec, respectively N 



N' , N" „ .., we shall have by means of equations (133), 



na' na 



2fJ!L h + N -?-, h'+N" r '^V'+N"~" l ,X+ &g. 

 na na na n a 



= \ jp^L +N*^y +N">™"„ 4-N'" 2 V f- ,„+ &c. I sin (at+l 

 \ na ' na' ' n'a" ' n"a ' j KJ ' ' 



In like manner the preceding values of I, V, I", Sec, will give, 



._ m m ,, AT „ m" in" „„ „ 



N- - l-\-N -r-fl 4-N „ -T4-N -*r»t + &c - 

 na 11 a na n a 



} (134) 



= 1 N m +N* m ', +*■■£ +N'"^, 

 [ na n a n a n a 



&c. > cos (gt-\-i3) 



(135) 



If the origin of the time t, be fixed at the epoch for which the values of h, I, 7b, T, 

 Sec, are supposed to be known; the two preceding equations will give, 



tan @ = 



__ in .. . - T , in ., , T „ in 7 „ . -, T „, in, ,,„ . „ 

 N— 7i4-N'- ,-, It 4- N"- „ Ji 4-N'"- „, -„-,7b"4-Sec 

 na na n a n a 



N — Z + N '-,— I + N -„—, T-\-N m -^-fff f-f&c. 

 wa na ji a n a 



(136) 



Since JV', iV", N'", &c. are given in terms of N, this indeterminate quantity will 

 disappear from the expression of tan /?. Having found /3, we may substitute it in 

 either of equations (134) or (135), and we shall obtain the value of N; and conse- 

 quently N, N", N", &c. will be known. We shall thus have the system of con- 

 stant quantities corresponding to the root g. Changing in the preceding equations 

 the root g successively into g x , g.,, g 3 , Sec, we shall obtain the values of the coastant 

 quantities, corresponding to each of these roots. 



5 September, 1S71. 



