352 LE VERRIER ON THE PERTURBATIONS OF PLANETS. 



To have regard to the equations (35.), I shall make 

 {i, -i+p)- {iy-i-p) =yp, 

 [h-i+p] + [i,-i-p] =tpi 

 these equations will then be written 



[i, — t\+yismnu + y^sin2nei + . . . + y^ sin j9 « a"^ ,„, 



n a. J • 



+ /i cosra« + #2 cos2 noc + . . . + tp cosp 

 and we shall easily deduce from them the values of [i, — i], 

 yp and tp. 



After these determinations we shall have 



{i,-i+p) = kxp + ^i/p, 



{i,-i-p)=^Xp-^yp, 



[h-i+p']=itp-^zp, 



li,-i—p]=^tp + ^Zp, 

 and thus all the coefficients of R wiU be calculated up to a given 

 decimal. 



17- We have already shown that the preceding method does 

 not permit any error to escape. We shsdl better appreciate the 

 advantages of it by observing that the arbitrary angle a. may be 

 adopted once for all ; and that thus it would be easy to deter- 

 mine beforehand the numerical values of the coefficients of the 

 formulce (24.), (26), ... so as to have no longer any but linear ex- 

 pressions to calculate, which is always very rapid. 



It is very important to observe, that in virtue of the excentri- 

 cities of the orbits the degree of convergence of the disturbing 

 function is different for the different values of the mean longi- 

 tude /' of the disturbing planet : so that there are great advan- 

 tages in not being obliged to employ in all the cases the same 

 number of values of the function R. We should not be able to 

 arrive at this, in the method whereby the circumference is divided 

 into equal parts, without being obliged to change incessantly 

 this division, which is impracticable. By the preceding calcu- 

 lations, on the contrary, we never employ more than the number 

 of the numerical values strictly necessary for each of the posi- 

 tions of the disturbing planet, without having to change the 

 formulae (24.), (27-), (29.), . . ., which we shall begin by esta- 

 blishing for the case in which the series is the least convergent. 

 There will afterwards be only to suppress one or more of the last 

 terms of these formulae in proportion as the calculation itself 

 will indicate the possibility of this, and taking care to neglect at 

 first the functions Ro and R^, then Rj and Rsi-i, and so forth. 



