INTRODUCTION. 



they are all increasing, (6) will be positive, and (4) will vary more slowly than (3); 

 but if they are all decreasing, (b) will be negative, and (4) will vary more slowly 

 than (3) only when (b) is numerically less than (a). Otherwise (3) will be pref- 

 erable to (4). But the reverse may, at the same time, be true with respect to the 

 convergency of these series. 



11. By a similar transformation of (4), putting 



bA — A = h"A , dA 1 — 8A = 8' i A l , hA 2 — 8A l = 8' i A. 2 , Sec, 

 we obtain 



(5) -fi^ = A + 8' A «' + fA t a* + PA, a 6 + • ■ ■ • . 



Continuing the same transformations and notation, we find 



(6) ^ = A + W,a ! + 8 1 i 1 «« + 8%a t + -, 



A + tfA,, a 2 + 8*Ai a 4 + & l A 2 a 6 + ' 



&c, &c. 



When we have a series corresponding to special values of s and i, that is, when 

 the coefficients in our series have been reduced to numbers, the number of such 

 transformations needed to obtain the series most desirable, both with respect to con- 

 vergency and the amount of variation relatively to a, will be a practical question 

 which can, in most cases, be settled by simple inspection. 



It has been found practicable to tabulate the following functions, 



>l 



log 



a D, b<-'2 



l02 T 



:h<$ 



log 

 a 2 n ft© 



a 2 D 2 5« 



a s a, b& 



log- 



l0 § „«-7 PR ' 



loff 



a 4 Dt tfP 



log 



a 2 6? 



for all values of i generally needed in the perturbative theories of the planets. 

 These tabular values have been limited to i = 9; since, for values of % larger than 9, 

 only few values of these, or any other functions, are needed. 



12. The following special case will show the effect of the above transformations. 

 In order to take the differences more readily, the series may be written vertically, 

 with the corresponding function at the top of the column. 



a 3 D iC«» 



a 8 D x blf 



fD.lff 



a 3 D^ b<f 



a 3 D % &< 10) 



a 3 D m b<-f 



a 12 



^"JJ 8 



a 8 (S 4 



cfi jS 6 



a*f 



a 2 r 



+ 567.35451 



+ 567.35451 



+ 567.35451 



+576.35451 



+576.35451 



+576.35451 



+ 1934.16309 



4- 1366.80858 



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+232.09956 



—335.25495 



—902.60946 



-|- 4442.53086 



4- 2508.36777 



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-(- 8494.50955 



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— 50.05895 



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-j- 6053.28610 



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— 4.30185 



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— 0.89955 



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— 0.03690 



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