INTRODUCTION. Vii 



Now it is evident that the series which gives the value of ° * * converges 



" a~ p lu 



more rapidly than any of the preceding ones ; and a little examination will show 

 that it also converges more rapidly than the one which will result from taking the 

 next order of differences. It is therefore the one best adapted to finding special 



values of a 3 D„ 6 ( l 0) ; but the function a - p * 6 s i s much the best one of the set to 



tabulate, since its whole variation, between the limits of a = and a = 0.75, is only 

 about half of that of the two functions which immediately precede and follow it. 



13. The series for computing those coefficients of which but few values can ever 

 be needed, are given on pages 42, 43, and 44. The coefficients corresponding to 

 such high values of i are mostly needed in the long-period terms in the theory of 

 Venus and the Earth. 



The few values of a 4 Dl bf needed in the theory of Jupiter and Saturn, were 

 computed from the series given by Mr. Walker. 



14. Resume (4), and write it 



(9) f(a) = A j-*? + /3 2 (8 A, a ; + 8 A, a i+2 + dA 2 a<+ 4 + 8A 3 a<+ 6 + ■■•■)• 



Now, since the series in this parenthesis has precisely the same form as (2), it may, 



by putting 



8A V — BA = 8 2 .4 , &A 2 — dA 1 =8 2 A u 8A 3 — BA t = 8 2 A. 2 , &c, 

 be written 



SA a*" 2 /3 2 + /3 2 (S 2 A a 1 + b 2 A l a i + 2 + S 2 A 2 a i + 4 + 8 2 A 3 a i + s + ' • • •) ; 



and, by substituting this form in (9), it becomes 



(10) f(a) = Aa'-V + SA a i - 2 ( 8 4 + /3 4 (8 2 A a* + 8 2 ^ a< + 2 + 8 2 4 a<+ 4 + ■ ■ ■ ■)• 



By a similar transformation 



(11) f( a ) = A a ; - 2 /3 2 + 84, a'' 2 ? + S 2 A a'" 2 (3 6 + /3 6 (8% a* + # A x c i + 2 + # A « i + 4 +' ' ' ') i 



and, continuing this process indefinitely, we finally get 



(12) f{a) = a ; - 2 f? (A + 8 A, /3 2 + 8 2 4 /3 4 + 8 3 A /3 6 + SM„ /3 8 + • • • • 8" 4 /3 2 "). 



In practice, the value of /(a) is obtained from the most convergent of these 

 transformed series, which may usually be determined by simple inspection. 



These are the forms adopted by Leverrier ; but we have derived them by a much 

 simpler process. 



15. Professor Peirce has also suggested transformations, which, as far as conver- 

 gency is concerned, leave nothing to be desired ; but, as most of his series are func- 

 tions of differences of the required quantities, tables constructed upon this basis 

 would not have been so practical as those we have given. Besides, it was very 

 desirable to make the whole subject rest upon a numerical basis prepared with such 

 great care as Mr. Walker's paper. 



The coefficients A , 8 A , S 2 A , B 3 A Q , &c, are the Leverrier Coefficients computed 

 by Mr. Walker. 



It will be seen that these coefficients are readily converted into those necessary to 

 compute the functions we have tabulated. For this reason, we have adopted the 

 series involving the same order of differences as those adopted by Leverrier ; al? 



