METHODS OF INTERPOLATION. 311 



111 the same way we can find the vahies of four, five, &c., coustants iu 

 the general forinuhi (12). For a series of the third order, tlie numbers 

 of terms iu the four groups are — 



=n, = -L^(\-cos^^=i{2- V-r 



?j2=w3=^]sr cos 4-=lN -v/u 



and the distances from the origin to the middle points of the groups are 



When these values are substituted in formula (8), the constants reduce to — 



A=^[(2 ^2-l)(S2+S3)-(Si+S4)J^ 



B=4[3(S3-S,)-(S,-Sx)] 

 C=^[(Si+S,)-(^/l>-l)(S,+ S3)] 



3>=^[(S4-Si)-(S3-S,)] 



For a series of the fourth order the numbers of terms in the five groups 

 are — 



ni=n,= lls('l - cos A=. 0954915 IvT 

 %=W4=iNf^ cos ~ —cos ^ j=iN 

 M3=N cos ^=.3090170 N 



and proceeding as in the case of formula (-lO), we find that the constants 

 are — 



A=:^[3.777709S3+i(Si+S5)-.411145G(S2+S4)] 

 B=i5[13.08S544{S4-S2)-^«-(S5-Si)] 

 C=^[55.33375(S2+S4)-71.73251 83-11^(81+85)]) (42) 

 r>=^[^^(S5-S:)-G3.28GG8(S4-S2)] 



E = ^[S3+(Sl + S5)-(S,+ S4)J 



"We might go on in the same way to find formulas for constructing series 

 of still higher orders. It will be noticed that in all these cases, in the 

 expression for the final constant, the sums Si, S2, &c., have the same 

 coefficient when taken without regard to sign, so that all the terms in a 

 given series will be of equal weight in determining the coefficient of the 

 highest power of x. 



