E 
488. ON CERTAIN FORMS OF INTERPOLATION. 
I EET. LEE 2—k.21+k 8—k 
(21) a- pa» [en HEP. p It xf: t : £ — pel 
Since the functions involved in (19) need not be restricted to integral values of i, 
similar expressions may be obtained for the differences between Y, and y; by sub- 
stituting mk for kin (9) and (20). By arranging these according to powers of mk 
and using their differences taken with respect to m, algebraic expressions may easily 
be got for any of the differences 97. To do this let m be substituted for y, in the 
series (4); and, p" m being one of the w^ order of differences of the new series, let 
9 (m) be the correspondingly placed difference of the series ym: then 
jd 9; (—m) — af (m) — wi (m), 
CR 
oi (m) = Of (m) + wi (mi) 
in which, putting i | 
(28) M Jn = HIEN HAN) = ixi wr = IA”, 
| a (m) = ibe ah oi a (4,— 44) + 00m. i36 "m fe, 
= = rm. ta (Go) E a im (CR rne). tn 
k ; 
| wi (m) = pem LS GE de Ah do) e mh AIR. s oe 
Formula (22) i is a perfectly general e connection between ‘the original and the interpolated 
differences of any order. 
When n=1, (22) gives the means of deriving all the first differences,* A. by 
computing the functions (24) for : or ia different values of m, according as i is 
even or odd, — in the latter case obtina 9, = 9, from (8). For this method the 
second form of 9 is adapted. But methods of this sort, however simplified in nu- 
merical application, seem unnecessarily laborious. | : 
Any interpolation may be made by combining (8) or (15) with (9), as follows. 
Compute the ð opposite each value of Y, and the ô or A opposite each interval. Then 
from the former fill in all the second differences, $2, and by their means the first differ- 
ences, 9,, and thence the function itself. If i is even, the two differences 9a and 
Fis, of which 0 is half the sum, must be derived from à' and the second difference, 
e, opposite it, by taking the sum and difference à'-r 1 us This method is some- 
times the most convenient, especially when ; — 5. If the higher differences are 
* For A. ebe of this sort, with: a different system of Geesen, see Mr. FERREL’S article on) 
Interpolation in the Mathematical Monthly, Vol. IL. p. 377. — 
we 
