THE ORBITS OF THE EIGHT PRINCIPAL PLANETS. 



33 



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

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

 and we shall therefore have 



m all 



na dt 



,m' dh^ 

 n'a' dt 



m 



ri'a" dt 



na 



n'a 



n a 



n a 



(132) 



If we substitute in this equation the preceding values of h, A', 7t", &c., Z, T, I", &c. , 

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



HIT. 



na 

 m 

 na 



na 



n a 



f + &o.=0 ; 



l + &c.=o ; 



&c. 



If we now multiply the preceding values of h, 7t', 7t", &c., respectively 



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

 n'a' n a 





h + N ' " 



(133) 



N-" 1 , 



na 



y (134) 



In like manner the preceding values of Z, Z', Z", &c., will give, 



> (135) 



If the origin of the time t, be fixed at the epoch for which the values of h, I, 7t', ?, 

 &c., are supposed to be known; the two preceding equations will give, 



m , 



n 



N 



tan 3 =- 



h + N' 7,' 

 na na 



N ' m rr 

 na 



, 

 na 



n a 



n a 



Z'"+&c. 



(136) 



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

 disappear from the expression of tan /3. Having found /?, 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^ g.,, g 3 , &c., we shall obtain the values of the constant 

 quantities, corresponding to each of these roots. 



5 September, 1871. 



