METHODS -OF INTERPOLATION. 



329 



Tablk hi.* 



Local tveifjhts of adjustment-formulas. 





5 



7 



9 



11 



13 



15 



17 



19 



21 



h 

 h 

 h 

 h 

 h 

 h 

 k 

 h 

 h 

 h 



.570 



.287 

 — . 072 



. 424 



.293 



.049 



— .054 



.344 



.270 

 .109 



— .016 



— .035 



.290 

 .245 

 .135 

 .020 



— .028 



— .023 



. 252 

 . 222 

 !l45 

 .056 



— .007 



— . 027 



— .015 



- 



222 

 202 

 140 

 076 

 015 

 018 

 022 

 010 



.200 

 .184 

 .143 

 .087 

 .034 



— .004 



— .020 



— .017 



— .007 



.182 

 .170 

 • . 137 

 .093 

 .047 

 .010 



- .012 



— .018 



- .013 



— .005 



.166 

 .157 

 . 132 

 .096 

 .057 

 . 022 



— .003 



— .015 



— .016 



— .010 



— .003 



c' 



£ 



.229 



.0822 



.0366 



.0137 



.0104 



. 00042 



. 00422 



. 00273 



. 00185 



* For some additions to this table, see Appendix. 



The upper line shows the number of terms included by each form- 

 ula ; the weights of the middle terms are found in the second line, 

 those of the terms next to the middle in the third line, and so on. 

 Tbe third decimal figure has in some cases been changed by a single 

 unit, so as to make the algebraic sum of all the weights in each formula 

 exactly unity. (Compare page 324 of the previous memoir.) The lowest 

 line of the table shows the ratio, for each formula, which the probable 

 error e' of a term in the adjusted series bears to the probable error s of 

 the corresponding term in the given series. These ratios are obtained 

 in the following manner: 



Let any adjustment- formula whatever, comprising 2 m + 1 terms, be 

 represented thus: 



Wo= ?o'«'o+ ?i(«i+ w_i) + 72(^2+ w_,) + .... + Z,,^ (M„^4- M_J (88) 

 where Uo is the middle term, '«i, Uo, &c., are the adjacent terms on the 

 right, «*_„ u__2, &c., are those on the left, and lo, Zi, h-, &c., are the local 

 weights in fractional form. Then, b^^ a process similar to that which 

 was followed in demonstrating formulas (G9) and (71), we shall find that 

 the fourth difference of any five consecutive terms in the adjusted series 

 is — 



J\ = (0 ?o - 8 /i + 2 Z2) «o + 1 7 ^- 4 (/o+ h) + I,} («i + w_0 . 



+ ^G l2-4:{h + h) + {k+h)\ ("2+ «-.) -f ] 



. . . + |0/_,-4 (/,,_,+ /_,) + (/..+ L-,)K''m-3+"M.n-2)) ' 



+ \ij I,^_,- 4 (L+Z,n-2)4-L-3|(Mn.-l + «-(nt-.)) 



+ (6 ?m-4?m-l4-C-2) (■«'m+*em) + (^m-l — 4 ?m)(^<m + l+ "-;m+l 

 + ?m(Min + 2+ W-(mf2)) 



ITow, take the series — 



h, 7i, k, k, h, ?3, Z., 0, 0, 0, 



and let its fourth differences be — 



(89) 



Jo, 



J. 



'm+l? 



