LE VERRIER ON THE PERTURBATIONS OF PLANETS, 349 



the preceding equations will be written 



and there will be deduced from them 



sin(i— 2) (2i + l)| sin (i— 2) (2 z— 1)^ 



^'-'"'^''■-' sm (i-2) « ^"'-^ sin(i-2)« ^ 



(31. 



cos(i-2) (2«+l)^ cos(i-2)(2i-l)| 



■t'!-2="^ i-2 = T- ci\ — "^(-2 ■ r- <vi ' 



sm(z— 2)« sm(z — 2)« 



And thus, rising from system to system, by these formulae, the 

 law of which is evident, we shall arrive by symmetrical calcu- 

 lations at the determination of all the coefficients Aj and B^, 

 A,_i and B,_i. . ., up to Aj and B„ A, and Bj in particular will be 

 given by the formulae of the ranks i and (i4- 1) of the system (10.). 



Lastly, the first of the equations (4.) will give B^ very simply. 



14. In short, the numerical operations which have to be ef- 

 fected to obtain a complete system of the values of A,- and Bj 

 corresponding to the same value of the longitude /', will be as 

 follows : — 



1°. The (2z+l) numerical values R^, Rj, Rg, . . ., and Rj, of 

 the disturbing function are determined. 



2°. By means of the formulae (9.) and (22.), the numerical va- 

 lues comprised in the table (16.) are determined. 



3°. By means of the formulae (24.), the 2 i quantities [i], and 

 [i\, [i — \\ and \i—l\, \} — '^s and [i—2'\^ up to [1]^ and 

 [l],+i, which only requires some logarithms, are calculated. 



4°. By means of the formulae (2/.), (30.) and those analogous 

 to them, the 2 (z — 1 ) quantities k'i_-^ and A'",_i, A',_2 and k"i_o, . . ., 

 up to Ar'j and k'\ are calculated. 



5°. The formulae (26.), (29.), (31.), and those analogous to 

 them, will give all the quantities A, and B; ; and the first of the 

 formulae (4.) will give the quantity Bq. 



All these calculations are symmetrical ; their nature admits of 

 executing them with exactness. We may moreover simply con- 

 trol the quantities (i)i, (i)2, {i).^, . . . For if we add all the equa- 

 tions which the formula (22.) gives, when we suppose that the 

 index k varies from n up to n', we shall have 



2:'(i+i),=2rx+2;;;,(i),-2cosi«2;::;(i)^.; . . (32.) 



