![{\displaystyle {\frac {\Delta y_{p}}{k}}={\frac {\operatorname {f} b-\operatorname {f} a}{b-a}}=\operatorname {f} _{1}(a,b),\ {\frac {\Delta ^{2}y_{p}}{k^{2}}}=2{\frac {\operatorname {f} _{1}(b,c)-\operatorname {f} _{1}(a,b)}{c-a}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c3f88f9a3d4cadca0e900f45cc4a4cf7a888739)
![{\displaystyle =2\operatorname {f} _{2}(a,b,c),\ {\frac {\Delta ^{3}y_{p}}{k^{3}}}=\ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/2719324bbbb77fe53da7cf71d5b54d921c81b749)
![{\displaystyle {\frac {\Delta y_{p+1}}{k}}={\frac {\operatorname {f} c-\operatorname {f} b}{c-b}}=\operatorname {f} _{1}(b,c),\ {\frac {\Delta ^{2}y_{p+1}}{k^{2}}}=2{\frac {\operatorname {f} _{1}(c,d)-\operatorname {f} _{1}(b,c)}{d-b}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5055fd9f315765666113382f42b4e4d0c20da4ac)
![{\displaystyle =2\operatorname {f} _{2}(b,c,d),\ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/b3bebfd970452302f2f2f69d57b47cea46b3d808)
![{\displaystyle {\frac {\Delta y_{p+3}}{k}}={\frac {\operatorname {f} d-\operatorname {f} c}{d-c}}=\operatorname {f} _{1}(c,d),\ldots \ldots \ldots \ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/cae568a1afb738ba5f412a7f46a4accbf3fe61ad)
. . . . . . . . . . . . . . . . . . . .
d’où, en substituant dans la formule (4),
![{\displaystyle y_{p+n}=y_{p}+{\frac {x-a}{1}}.{\frac {\Delta y_{p}}{k}}+{\frac {x-a}{1}}.{\frac {x-b}{2}}.{\frac {\Delta ^{2}y_{p}}{k^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/11ea81e3be31604047dbf011ab8cfcaa36afa421)
![{\displaystyle +{\frac {x-a}{1}}.{\frac {x-b}{2}}.{\frac {x-c}{3}}.{\frac {\Delta ^{3}y_{p}}{k^{3}}}+\ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/2a138cb105530ca7e62435f71b88844f9d7a57b1)
(6)
Si nous posons
nous aurons
![{\displaystyle {\begin{aligned}&x-b=(x-a)-(b-a)=nk-k=(n-1)k,\\&x-c=(x-a)-(c-a)=nk-2k=(n-2)k,\\&\ldots \ldots \ldots \ldots \ldots \ldots \ldots ,\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c89e2a1ca811ce13ed726fa2be876d8fe7729df3)
de sorte qu’alors cette formule deviendra
![{\displaystyle y_{p+n}=y_{p}+{\frac {n}{1}}.\Delta y_{p}+{\frac {n}{1}}.{\frac {n-1}{2}}.\Delta ^{2}y_{p}+{\frac {n}{1}}.{\frac {n-1}{2}}.{\frac {n-2}{3}}.\Delta ^{3}y_{p}+\ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/82d388b39a682bd2f955960329e257b6906403d7)
(7)
et, par suite, en supposant
nul,
![{\displaystyle y_{n}=y+{\frac {n}{1}}.\Delta y+{\frac {n}{1}}.{\frac {n-1}{2}}.\Delta ^{2}y+{\frac {n}{1}}.{\frac {n-1}{2}}.{\frac {n-2}{3}}.\Delta ^{3}y+\ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/693f910d5f578a883333607c3a25f3559867b0b7)
(8)
ce sont les formules connues d’interpolation, pour le cas de l’équidifférence des valeurs de la variable indépendante.