436 ON CERTAIN FORMS OF INTERPOLATION. 
Let i be a positive integer, and k = d then after the insertion between Y, and 
Y,,,of i—1 new values of the function (which is called interpolating to k*), 
let a portion of the series and its differences be 
E = 
> E Bm 
9, ep The quantity 9 wil be an implicit function of f, the . 
Y $i . i s i : E 
9% i. values indicated in (4) corresponding to"t= f, — $4, 
` is $i ttik, i + k, Sc.; let its value for ? — LA be 
4 ey a 0 : Ys 
(4) . ô, 62 will then, i» all cases, whether i is odd or even, be 
B i. the same function of à , and its differences that ô, is of 
2 ` . D 
y hes 9 Y, and its differences; that is, if 
i- i=} ' i 
1 ài 
(uc CN 
| Yin 
iius dt: = exi (XN), e : 
SG = p (0-1) = 9-9—,(Y) = (KH) 
(5) 4 and similarly for similar functions of higher orders: 
Ve, = 94, 09 = 944 9 9-4 (Lo) = 9 (NM), 
A = 9 OLy) = - 9-4 P P (P) — 94 (Lo), &e.; 
q" representing the result of the » successive functional operations indicated in each 
third member. : 
From the well-known development 
© & —5 than + OEY gt EAS y, y EEG D GT 44 be, 
is directly obtained 
La ah op 
7 = =| Em e ] 
(7) Pt ER Cd eee en TA A ET eS ere 5 fti &c. |, 
which is a linear function of certain differences, which are themselves linear functions ; 
therefore q” is obtained by n symbolical multiplications of the series (7) according 
to the law (3). The subscript numbers are seen to result so that, omitting them from 
the operation, we have à" = the symbolical n^ power of 8; that is 
1— 
24 
(8) a + [a tt tt Pal, 
