324 METHODS OF INTERPOLATION 



n, = ne = *N ^cos ^ - cos ^^ = .1631759112 N 



M5 = N cos ~= .1736481777 N 



A = 4 ■! 6.718900297 S5 + .238136413 (S3 + S,) + J (Si + Sg) 



-.650752476 (S4+ Se) - .140549851 (S2+ Ss) \ 

 B = i. j 40.1146355 (S6-S4) +19.9343410 (S8-S2) -25.6631423 (S.-Sa) 



C = i- I 283.1606672 (S4-+ So) + 67.0638653 (S2 + S3} - 409.8722695 S5 



- 111.5059994 (S3 + S7) - ^^^ (Si + S9) \ 



D = 1 1 1053.695090 (S, - S3) +^^(89 - Sj) - 923.898793 (Sg - S2) 



- 823.638243 (SG-S4) | 



E = i, ! 6330.21204 S5 + 3859.01970 (S3+ S,) +^ (Si + So) 



- 5527.64697 (S4 + Sg) - 2594.89308 (S2 + Sc) | 



F = ^^ • 9605.23187 (83 - S^) + 5158.73566 (So- S4) -^JJ— (S<, - Si) 



- 8494.59768 (S- - S3) \ 



G = ^, I 33866.1397 (S4 + So) + 25281.7661 (S2 + S^) - 35452.4678 S5 



200704 ) 



-29849.4159 (83 + 8^) — g— (S1+S9) } 



H = 4 i ^^TT^ (S9 - Si) + 19011.3716 ( S, - S3) - 25613.9516 (So - S.) 



- 10115.7395 (S6-S4) I 



I = ^1^ I S5+ (S3+ S,) + (Si+ Sg) - (S4+ Sg) ~ (S2+ So) } ■ 



We have now tlie lueans of representing any given series by an equa- 

 tion of a degree not higher tban the eighth, assuming groups of either 

 equal or unequal exteut. In constructiiig the graduated series from 

 such an equation, it will be most convenient to proceed as stated on 

 page 329 of the previous memoir. The work of finding the values of 

 C'.r^, B'x\ &c., will be much focilitated by using the accompanying 

 table, which shows at once the seveu-figure logarithms of the powers of 

 all the values of x which can be required in constructing a series of not 

 more than one hundred terms. Increased accuracy, too, will be attained 



