320 METHODS OF INTERPOLATION. 



The expressions for the alternate constants B and D cannot be put to 

 the same test, because (S4— S2) and (S5— Si) are necessarily zero; so we 

 will make another supposition, namely, that the given series consists o.^* 

 the natural numbers, the middle term being 0, and the successive terms 

 on the right being 1, 2, 3, &c., while those on the left are —1, —2, —3, 

 &c. Here we ought to have, in the equation of the series, B = 1, and 

 all the other constants equal to 0. The sums of the terms in the several 

 groups are- — 



S, = 





S.= 



Hence we have 

 B 



A I 13.088544 n, {n, + n,) - ^ n, (2 n, + W3+ n,) | 



D =i-^ I ^ % (2 01, + «3 + n,) - 63.28668 01, {n, + n,) J , 

 and substituting the values of 713, n^, and %, we jQnd — 



B = 1.00000007 D = ^2 (- .0000003) 



which are very nearly unity and zero, as they ought to be. 



The accuracy of each of the following formulas can be tested in a 

 similar way. The five-group formula (42), when the decimals are car- 

 ried out a little further, becomes — 



Formula (79). 

 wi = ws = i N ('1 _ COS0 = .0954915028 N 

 «3 = "4 = ^- N r COS g — cos ^ J = 1 N 



9 - 



^.3 = ]sr cos "p-= .3090109944 N 



5 



A = 1 { 3.777708764 S3 + i (Si + S5) - .411145018 (S2 + S4) \ 

 B = A J 13.08854382 (S4 - S^) - ^| (S5 - Si) | 

 C = i-3 1 55.3337474 (S2 + S4) - 71.7325052 S3 ™ ^ (Si + S5) \ 

 D = i, I ^ (S5 - Si) - 63.2866805 (S4 - S2) \ 

 E = ^ I S3 + (Si + S5) - (S2 + S4) J 

 The six-group formula can be expressed without decimals : 



