340 LE VERRIER ON THE PERTURBATIONS OF PLANETS. 



6, We shall be able to eliminate o,, then b^ between these 

 equations. But the relations which would contain in general b^ 

 without aii would be wanting in symmetry, and we shall make no 

 use of them. It is preferable to ehminate at once a^ and b^, and 

 to form (2i — 2) new equations which will be deduced from the 

 preceding combined three by three as follows. 



Let us designate by Mi, Mg and M3 the first members of three 

 of the equations (10.) taken consecutively. On making the com- 

 bination 



Mi + M3-M2(^ + 4?->), (11.) 



of these quantities, the terms in «j and b-^ will disappear. Any one 

 may convince himself of this by substituting in this formula, in 

 place of M,, Mg and Mg, the following expressions of the parts 

 dependent on a, and on b^ in the first members of three of the 

 consecutive equations 



(flj^* —b^ a?-''-') {\—x), 

 (ai^*+'-6, ^-''-2) [\—x), 

 (a]«*+2-i, j:-'^-^) (1-^). 

 Let us take three other corresponding terms in the first mem- 

 bers of the same equations ; for example the terms, 

 {ciiX''' — ^>ir-«*+i)] (!—.«•% 

 [a.a?'(*+i'-di ^>-. (fr+a)] (i_.r'), 

 \ai ^'(*+2)_i. ^-'(*+3)] (1 -X''). 



By submitting them to the combination (11.), it will easily be 

 found that expressions of the following form will result, 



[«,.^'*-5,;r-'"(*+="] (l-a^'+') (l-a7") (1-^'-'), 



and this formula will serve to construct the first members of 

 the new equations by giving to the index i values from 2 up to i, 

 and to the quantity k values from up to (2 i— 3). 



With respect to the second members of these new equations, 

 we shall designate their real parts by (2)i, (2)2, (2)3, . . ., and by 

 remarking that according to the first of the conditions (6.), 



A-^ + A'~' = 2C0S«, 



we shall find 



(2)i = (l), + (l)3-(l)2 2cos«,1 

 (2)2=(l)2+(l)4-(l)32cos«, 



(2)3=(1)3+(1)5-(1)4 2C0S«, 

 &c 



(12.) 



