METHODS OF INTERPOLATION. 34T 



But if the number of groups assumed is odd, we have the m + 1 equa- 

 tions — 



A,(Si-A%) + A2(S2-Awi)+. .+A^(S„, -Awi) + (S^+i-A%)=:0- 

 Ai(S2-A%) + A2(S3-A%i)+. .+A^(Sni+i-A%i) + Sm+2-A«i)=0, 



Ai(Sm+i-AMi)+A2(Sm+2— Awi)+. .+A4S2m— AWl)+(S2ra+l— A»i])=0, 

 and Awi being eliminated by subtracting each equation from the suc- 

 ceeding one, there will be m equations remaining from which the values 

 of the scale-terms may be found as before. Whether the number of 

 groups assumed is 2 m or 2 m + 1, the equation of relation will be — 



z'^ -f A„, ;2"-i + A„_iS'"-2 + + Aa + Ai ^ 



This numerical equation of the mth degree being solved, its real roots 

 are the values of the constants fi^\ /Si^', /Sg^, &c. As for the imaginary 

 roots, each pair of them are the roots of a quadratic factor of the equa- 

 tion of relation. These factors being found, let us denote them by — 



s^ -f » ^ -{- g, =0 

 «^+i?i« + 2i = 

 &c., &c. 

 Then we shall have — 



q =f^ \ p = —2r'^ cos (7i d) \ 



qi = n'"^ [ i^i = - 2 n'^ cos [h d,) ^ (105) 



&c.,&c. ) &C., «&C. 1 



and thus the values of the constants y, yi, y2, &c., and of the constant 

 arcs 0, 01, 0-2, &c., become known. Substituting them and the values of 

 /5, '^i, ^2, &c., in the general formula (102), and assigning to n the numerical 

 value of Uij and to S the successive numerical values of Si, S2, &c., and 

 to X the corresponding values for the middle points of the grouj)s, we 

 shall have a system of equations which, besides the constants A, B, Bi, 

 B2, &c., 0, Oi, O2, &c., and D, Di, D2, &c., contains only numerical quanti- 

 ties. These constants will be m or m -f 1 in number, according as there 

 are 2 m or 2 m + 1 assumed groups. We shall only have to form as 

 many equations as there are constants, and then the values of the con- 

 stants can always be found. 



Having thus completed the determination of all the constants in 

 formula (102), we are enabled to interpolate the sum S of any group of 

 ji terms in the graduated series; and if we take — 



n = l, S = w 



the equation of the series may be reduced to the simple form — 



« = A + &/3^+&i/?f 4-&2/52^+ &c., ^ 



-\-\c sin {xO) -{- dcos {x 0)\ y'' \cif)a\ 



+ \ci sin {x 0{) + di cos [x Oi) \ yf C 



+ &c., «&c. J 



The sums of the terms in the 2m or 2m-\-l assumed groups will be the 



same as in the given series. If the number of groups assumed was 2??/, 



the graduated series will be recurrent, and of the mth order; if the 



