ON CERTAIN FORMS OF INTERPOLATION. ` 439 
very large, the 93s must themselves be interpolated, either by a Bii ¿perbipo, 
or by the use of (10) or (16) and (iis <: J TUE 
An objection to these processes, when many STEE iste are to be iüéerted, 
is that the sum of the computed first differences will not generally be exactly equal 
to the differences between two successive values of Y ; nor will the sum of the com- 
puted second differences exactly fill in the computed first differences, etc. These 
difficulties have to be remedied by subsequent arbitrary correction, or by com- 
puting the higher differences to one or more decimal places beyond those of Y. 
For this reason, interpolation to fourths, sixths, ninths, &c. should generally be 
done by successive interpolations to halves and thirds; and the choice of the 
special method of doing this must depend more or less upon the particular charac- 
ter of the function to be interpolated. The following are some practical methods 
with the numerical coefficients in the various series required. 
INTERPOLATION TO Harvrs. 
(21^) On = $[4 — $60 + s dx +1, 
(57) — ð —14, 
(27) y —i1[4—X$34o0 344 ...]. 
The interpolated differences may be obtained in’ successive pairs by taking the 
hal£sum and half-difference of the quantities ` 
rc pl i= #24, babe d 
25 BRNS 
i 9% = of or = 97. 
When 9? can readily be obtained by interpolating the djs, it may be more con- 
venient to use (9") instead of (21") to obtain it. 
 IwTERPOLATION To THIRDS: 
p^. 04, = Bai hr de ei 
e EH AAPA RA] c 
(211) 
(8") 6 Stl Sao Aa 
(9") e = 4H BG]. 
D This general method, as far. as- relates to odd values of i; with algebraic forms for j^^! and Jj” upto | 
A is given by Ewckz in the Astronomische Nachrichten, Band 29, No. 695. Also a partial statement, 
obtained by induction apparently, of the symbolic connection of the series (8) .. . (12). 
