ON CERTAIN FORMS OF INTERPOLATION. . AB 
(9) c pep. Aq UB RE... 
(10) patie o4 te, E), 
(11) st = k P at LIE TAM A... 
CH o =P 14-215 4 ...]. 
It will be observed that the notation 07^ was substituted in (4) for each difference 
of the form 975, because the symbolic even powers of à as there defined arè always 
identical with 9?*. When i is odd, 9?"+1 is also identical with the middle 9?"*! þe- 
tween Y, and Y;; and all the series (8)...(12) may be used directly. When i is 
even, let à?"*! denote half the sum of the two Ó?"*!s next above and below the 
middle point between Y, and Y,; then in each case it will be the same function of 
82^ and its differences, that Y is of Y, and its differences; that is, if | 
(13) y = (Xj, | then Q^" — oi (ain). 
But we have for these functions of k, whether 1 i is odd or even, 
(14) à (b) = 45 (25); that is, g' (E) = 4p (2k); 
therefore the series for à?"*! may be obtained by substituting 2k for k in (7) and 
symbolically multiplying this into the series for 1 6°". This gives 
db: Te Av oa gi] 
(16) | DEM, (= idia um) Y O 
(17) parpi 4 ...]. 
Besides the symmetrical relations obtained above by introducing the functions 3, 
the following may be obtained directly from (6): — 
(18) — t=O — 6€, $5, = 06, + ei 
h+ 924 — 4+ 03, 
19 
in which 
1— ]—p PP 
(20) 844 =F [24 — Ex x T Aud ES 15 Ly, — to], 
VOL. VIII. o1 
