350 LE VERRIER ON THE PERTURBATIONS OF PLANETS. 



the sign 2 being relative to the different vaUxes of index k. This 

 relation will permit of verifying rapidly the whole series of the 

 coefficients {t)^ or only a certain number among them taken con- 

 secutively, which will easily disclose the errors that may have 

 shpped in. 



Lastly, it will be necessary for the first and second of the 

 equations (4.) to agree in giving the same value of the constant 



Bo. 



15. Let us return to the relations (3.), and first to the first 

 among them. By the preceding calculations we shall be able to 

 determine the values of Bq corresponding to the mean longi- 

 tudes 



/' = 0, Z' = «, /' = 2 «,..., I' =2 ice, 



which will furnish (2i + l) relations to determine the constant 

 C of the disturbing function, and the coefficients (0, i') and 

 [0, i'], corresponding to the different values of i' from 1 up to i. 

 It is clear in fact that the index i being null, it suffices to attri- 

 bute to i' positive values. The relations to be solved being more- 

 over wholly similar to the relations (4.), we have nothing to add 

 on this point. 



16. It remains for us to solve the system of the two last of the 

 equations (3.), as it is presented when we give to /' the values 



yj, 1, ^, . . ,, ^ I. 



Restricting ourselves to the system composed of the equa- 

 tions furnished by one only of the equations (3.), it would be 

 still more completely similar to the system (4.), and it would be 

 treated in the same manner. But we should be obliged to em- 

 ploy thus twice as many numerical values of the function R as 

 when using the two equations (3.) at a time. We shall then at- 

 tempt to employ them, and shall find in this the advantage of 

 decomposing the equations which afford the coefficients [i, i') 

 and [i, i'] for one value of i and for different values of i', positive 

 or negative, in two separate systems. 



I remark witli this view, that in giving to i' the value —i, the 

 coefficients {i, i') and [i, i'] become of the order zero in relation 

 to the excentricities and the inclinations ; that to have regard to 

 the coefficients which are of the order p in relation to these ele- 

 ments, it will be necessary to give to i' all the values from 

 { — i—p) up to {—i+p)- And thus the system of the two equa- 

 tions (3.) will become for one value of I' equal to nx : 



