Mais en vertu du théorème des fonctions homogènes, on a
![{\displaystyle {\frac {\operatorname {d} u'_{n}}{\operatorname {d} x'}}x'+{\frac {\operatorname {d} u'_{n}}{\operatorname {d} y'}}y'=nu'_{n},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bdb35830a6a7f7ba8738c53e8b5af0a8e435380e)
![{\displaystyle {\frac {\operatorname {d} ^{2}u'_{n}}{\operatorname {d} x'^{2}}}x'^{2}+2{\frac {\operatorname {d} ^{2}u'_{n}}{\operatorname {d} x'\operatorname {d} y'}}x'y'+{\frac {d^{2}u'_{n}}{\operatorname {d} y'^{2}}}y'^{2}=n(n-1)u'_{n},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bc6e8a1ebe60e6172b1172ffbcc7edad4dc8fac6)
![{\displaystyle {\frac {\operatorname {d} ^{3}u'_{n}}{\operatorname {d} x'^{3}}}x'^{3}+3{\frac {\operatorname {d} ^{3}u'_{n}}{\operatorname {d} x'^{2}\operatorname {d} y'}}x'^{2}y'+3{\frac {\operatorname {d} ^{3}u'_{n}}{\operatorname {d} x'\operatorname {d} y'^{2}}}x'y'^{2}+{\frac {d^{3}u'_{n}}{\operatorname {d} y'^{3}}}y'^{3}=n(n-1)(n-2)u'_{n},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3079db1aafcf7f9f91f07e9b4d4d3332506bd754)
![{\displaystyle \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ca668a623e6e4247c8bf1592b3eae19ca325134)
![{\displaystyle {\frac {\operatorname {d} ^{n}u'_{n}}{\operatorname {d} x'^{n}}}x'^{n}+{\frac {n}{1}}{\frac {\operatorname {d} ^{n}u'_{n}}{\operatorname {d} x'^{n-1}\operatorname {d} y'}}x'^{n-1}y'+{\frac {n}{1}}.{\frac {n-1}{2}}{\frac {\operatorname {d} ^{n}u'_{n}}{\operatorname {d} x'^{n-2}\operatorname {d} y'^{2}}}x'^{n-2}y'^{2}+}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c979160162109fde0db92f696bf665a66592c303)
![{\displaystyle \ldots +{\frac {\operatorname {d} ^{n}u'_{n}}{\operatorname {d} y'^{n}}}y'^{n}=1.2.3\ldots nu'_{n}\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5bcb0b4a26c52212dfa3f94c85b3c18d795006f5)
En substituant donc, il viendra
![{\displaystyle t_{n}=u'_{n}\left\{m-(m-1){\frac {a}{1}}+(m-2){\frac {n}{1}}.{\frac {n-1}{2}}\right.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5d9eb2df057eaa2d50057efd475ea6f934f613d6)
![{\displaystyle \left.-(m-3){\frac {n}{1}}.{\frac {n-1}{2}}.{\frac {n-2}{3}}+\ldots \pm (m-n)\right\}\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7a3ce9f67c7ab8b52bd0ae43eab6277a7526761f)
c’est-à-dire,
![{\displaystyle t_{n}=mu'_{n}\left(1-{\frac {n}{1}}+{\frac {n}{1}}.{\frac {n-1}{2}}-{\frac {n}{1}}.{\frac {n-1}{2}}.{\frac {n-2}{3}}+\ldots \pm 1\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9658ed297bfb8ced159cc0d3b8951ffdd7f85562)
![{\displaystyle +nu'_{n}\left(1-{\frac {n-1}{1}}+{\frac {n-1}{1}}.{\frac {n-2}{2}}-{\frac {n-1}{1}}.{\frac {n-2}{2}}.{\frac {n-3}{3}}+\ldots \mp 1\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2bb61236c1ae2f24975a2cb95dee1016ccd29fd3)
ou bien encore
![{\displaystyle t_{n}=u'_{n}\left\{m(1-1)^{n}+n(1-1)^{n-1}\right\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/64a8b63b6ba2300d63cfb23d6152759751bb5ba3)
Pour toutes les valeurs de
plus grande que l’unité, cette quantité devient nulle ; pour
elle se réduit à
et, comme
est une constante, pour
elle se réduit à
Const. On a donc