16
INTÉGRATION
![{\displaystyle B=Az,\qquad C={\frac {3z^{2}}{6-z^{2}}}A,\qquad D={\frac {z^{3}}{6-z^{2}}}A\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b3b248f38fed3cc035139d3774e78390729ae676)
mais, par la condition qui détermine la constante, on a
![{\displaystyle 1=A-B{\frac {a}{z}}+C{\frac {a^{2}}{z^{2}}}-D{\frac {a^{3}}{z^{3}}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/303d35b38f5c8095b2e24c4c3caeb393dca44ff1)
en substituant donc, il viendra
![{\displaystyle 1-A\left\{1-a+{\frac {3a^{2}}{6-z^{2}}}-{\frac {a^{3}}{6-z^{2}}}\right\}\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ba8831f4c825fd0cc4757a19f17e487ecb69f246)
faisant enfin
et changeant
en
il viendra, pour seconde approximation
![{\displaystyle e^{x}={\frac {1}{1-{\frac {x}{1}}+{\frac {x^{2}}{2}}-{\frac {x^{3}}{6}}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/58a1e6aa1de83f46701d1a7f714397c1156df70a)
Posons encore
![{\displaystyle y=A+Bx+Cx^{2}+Dx^{3}+Ex^{4}+Fx^{5}\,;\qquad }](https://wikimedia.org/api/rest_v1/media/math/render/svg/f7e8612669eb0d2165edfbe3dea312f8e32517d7)
(5″)
d’où
![{\displaystyle {\frac {\operatorname {d} y}{\operatorname {d} x}}=B+2Cx+3Dx^{2}+4Ex^{3}+5Fx^{4},\qquad }](https://wikimedia.org/api/rest_v1/media/math/render/svg/6cbd95f7f03b4974e329965f9f39849f5b9a64a1)
(6″)
substituant ces valeurs dans l’équation (1), elle deviendra
![{\displaystyle z\left(A+Bx+Cx^{2}+Dx^{3}+Ex^{4}+Fx^{5}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c0dd5709e3e9ed8f146eef76af5dc185e2b29aed)
![{\displaystyle -\left(B+2Cx+3Dx^{2}+4Ex^{3}+5Fx^{4}\right)=0\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/23459b4bb20b9a9276f071ad4a70f4e49a196005)
ou en ordonnant et posant, pour abréger,
![{\displaystyle {\begin{alignedat}{2}Az-\ \ B=&A',\qquad &Dz-4E=&D',\\Bz-2C=&B',&Ez+5F=&E',\\Cz-3D=&C',&Fz=&F'\,;\end{alignedat}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/83e2c1b8f74e1c86c4753a9bab0686eccf4868e6)