346 METHODS OF INTERPOLATION. 



2/= A + B (log' 13) ,3^ + Bi (log' A) A^ + B^ (log' (3,) [i^^ + &c. 



+ {(C log' r - D^) sin {X d) + (D log' y + G 0) cos [x 0) \ f „^ 

 ■ +KOilog'n-D36'i)sm(a^'^i)+(Dilog'n+Oi^i)cos(£c^i)|n^^^ 



+ &c., &c. 



By integration we get — 



fydx = Kx^^^-\- Bi ^-^ + B2 ;V + &c. 

 + 1 sin (a? (?) -f- D cos (a? (9) ^ ^^ 

 + jCi sin (ip^i) + Di cos (a?6'i)(^i== 

 + {C2 sin {x 0.2) + D2 cos {x 02)1x2' 

 4- «&c., &c., + constant; 



and taking this between the limits x — ^n and x -{- ^n, and employing Ic, 

 Ici, /i-2, &c., and Z, ^1, k, &c., as auxiliary letters, we have — 



/,- = {yhn — y-in) COS ^ {U 0) ' ■% I z= (j'i» + y-^i) SlU ^^ (U 0) \ -^ 



h = in^'' - ri~^'0 COS ^ (% ^i) V li = (n4« + ri-"») sin ^ (w ^i) ( j 



&c., &c. ) &c., &c. ) f 



S r= A% + B (/?*» - 13-hn) fr + Bi (,AA» - ft-4-«) /5i^ + &c. > (102) 



+ {(0 A; — D Z) sin (a? 6) + (D /t + C Z) cos [x 0)\f I 



+ |(Ci^-i-DiZi)sin(a;6'i) + (Di/.i + Cili)cos(a?^i)}j'i- ] 

 + &c., &c. / 



This is the expression for the sum S of any n terms taken in a group, 

 the abscissa of the middle point of the group being x. It is of the same 

 form as the expression for y in (101), so far as the variable x is concerned. 

 To apply it to the graduation of a given irregular series, we assume 

 certain groups of equal extent and equidistant, and denote the number 

 of terms in each group by %i, and the constant interval between the 

 middle points of the groups by li. If we substitute n^ for n in (102), and 

 assign to x in succession the values corresponding to the middle points 

 of the assumed groups, we shall obtain expressions for the sums Si, S2, 

 S3, &c., of the several groups. These sums will form a recurring series 

 of the mth order if the number of groups assumed is the even number 

 2 w?, in which case the constant A is to be omitted from the function 

 altogether. But if an odd number, 2 m + 1, of groups is assumed, then 

 A must be retained, and the differences — 



(Si -Aw,), (S2 — Awi), (S3— Awi), &c., 

 will form a recurring series of the mth order. Let the m terms of the 

 scale of relation be denoted by — 



— Ai, — A2, - A3, — A,!, 



When the number of groups assumed is even, the numerical values of 

 the scale-terms Ai, A2, &c., are found irom the m equations — 

 Ai Si + A2 S3 + A3 S3 +......+ A,, S„ + S^+i = ^ 



Ai S2 + A2 S3 + A3 S4 + + Ao, S^+i + S^+2 = ^ 



Ai Sm+ A2 Su,+, + A3 Sm + 2 + + A]^ S2m-1 + S2m = 0. 



