Страница:Лахтинъ Л. К. Алгебраическiя уравненiя, разрѣшимыя въ гипергеометрическихъ функцiяхъ.pdf/261

Положивъ въ уравненіяхъ (57):

${\frac {\eta _{1}}{\eta _{2}}}=u,$

‎находимъ:

$\left.{\begin{array}{l}u^{8}+14u^{4}+1={\dfrac {(R-c)^{2}R^{3}}{c'{}^{2}R'{}^{4}\eta _{2}^{8}}},\\u^{12}-33u^{8}-33u^{4}+1={\dfrac {(R-c)^{3}R^{4}}{c'{}^{3}{\sqrt {c'}}R'{}^{6}\eta _{2}^{12}}}{\sqrt {R-c}}.\end{array}}\right\}$

(58)

‎Исключивъ $u$ изъ уравненій (58), находимъ:

${\begin{matrix}\eta _{2}^{24}-{\dfrac {5}{2\,.\,3}}{\dfrac {(R-c)^{2}R^{3}}{c'{}^{2}R'{}^{4}}}\eta _{2}^{16}-{\dfrac {5}{2^{4}}}{\dfrac {(R-c)^{3}R^{4}}{c'{}^{3}{\sqrt {c'}}R'{}^{6}}}{\sqrt {R-c}}\,\eta _{2}^{12}\,-\\-\,{\dfrac {3\,.\,5}{2^{8}}}{\dfrac {(R-c)^{4}R^{6}}{c'{}^{4}R'{}^{8}}}\eta _{2}^{8}-{\dfrac {1}{2^{8}}}{\dfrac {(R-c)^{5}R^{7}}{c'{}^{5}{\sqrt {c'}}R'{}^{10}}}{\sqrt {R-c}}\,\eta _{2}^{4}\,-\\-\,{\dfrac {1}{2^{10}\,.\,3^{3}}}{\dfrac {(R-c)^{6}R^{8}}{c'{}^{6}R'{}^{12}}}\left({\dfrac {R-c}{c'}}-R\right)=0.\end{matrix}}$

(59)

‎или, освобождая это уравненіе отъ радикаловъ и опустивъ индексъ 2 при $\eta _{2}$:

${\begin{matrix}\left\{\eta ^{24}-{\dfrac {5}{2\,.\,3}}{\dfrac {(R-c)^{2}R^{3}}{c'{}^{2}R'{}^{4}}}\eta ^{16}-{\dfrac {3\,.\,5}{2^{8}}}{\dfrac {(R-c)^{4}R^{6}}{c'{}^{4}R'{}^{8}}}\eta ^{8}\,-\right.\\\left.-\,{\dfrac {1}{2^{10}\,.\,3^{3}}}{\dfrac {(R-c)^{6}R^{8}}{c'{}^{6}R'{}^{12}}}\left({\dfrac {R-c}{c'}}-R\right)\right\}^{2}\,-\\-\,{\dfrac {5^{2}}{2^{8}}}\left\{\eta ^{8}-{\dfrac {1}{2^{4}\,.\,5}}{\dfrac {(R-c)^{2}R^{3}}{c'{}^{2}R'{}^{4}}}\right\}^{2}{\dfrac {(R-c)^{7}R^{8}}{c'{}^{7}R'{}^{12}}}\eta ^{8}=0.\end{matrix}}$

(60)

‎Таково уравненіе, соотвѣтствующее октаэдрической формѣ $f(\eta _{1},\;\eta _{2})$.

Подставивъ въ него значенія постоянныхъ:

$c=108,\ c'=1,$

Тот же текст в современной орфографии

Положив в уравнениях (57):

${\frac {\eta _{1}}{\eta _{2}}}=u,$

‎находим:

$\left.{\begin{array}{l}u^{8}+14u^{4}+1={\dfrac {(R-c)^{2}R^{3}}{c'{}^{2}R'{}^{4}\eta _{2}^{8}}},\\u^{12}-33u^{8}-33u^{4}+1={\dfrac {(R-c)^{3}R^{4}}{c'{}^{3}{\sqrt {c'}}R'{}^{6}\eta _{2}^{12}}}{\sqrt {R-c}}.\end{array}}\right\}$

(58)

‎Исключив $u$ из уравнений (58), находим:

${\begin{matrix}\eta _{2}^{24}-{\dfrac {5}{2\cdot 3}}{\dfrac {(R-c)^{2}R^{3}}{c'{}^{2}R'{}^{4}}}\eta _{2}^{16}-{\dfrac {5}{2^{4}}}{\dfrac {(R-c)^{3}R^{4}}{c'{}^{3}{\sqrt {c'}}R'{}^{6}}}{\sqrt {R-c}}\,\eta _{2}^{12}\,-\\-\,{\dfrac {3\cdot 5}{2^{8}}}{\dfrac {(R-c)^{4}R^{6}}{c'{}^{4}R'{}^{8}}}\eta _{2}^{8}-{\dfrac {1}{2^{8}}}{\dfrac {(R-c)^{5}R^{7}}{c'{}^{5}{\sqrt {c'}}R'{}^{10}}}{\sqrt {R-c}}\,\eta _{2}^{4}\,-\\-\,{\dfrac {1}{2^{10}\cdot 3^{3}}}{\dfrac {(R-c)^{6}R^{8}}{c'{}^{6}R'{}^{12}}}\left({\dfrac {R-c}{c'}}-R\right)=0.\end{matrix}}$

(59)

‎или, освобождая это уравнение от радикалов и опустив индекс 2 при $\eta _{2}$:

${\begin{matrix}\left\{\eta ^{24}-{\dfrac {5}{2\cdot 3}}{\dfrac {(R-c)^{2}R^{3}}{c'{}^{2}R'{}^{4}}}\eta ^{16}-{\dfrac {3\cdot 5}{2^{8}}}{\dfrac {(R-c)^{4}R^{6}}{c'{}^{4}R'{}^{8}}}\eta ^{8}\,-\right.\\\left.-\,{\dfrac {1}{2^{10}\cdot 3^{3}}}{\dfrac {(R-c)^{6}R^{8}}{c'{}^{6}R'{}^{12}}}\left({\dfrac {R-c}{c'}}-R\right)\right\}^{2}\,-\\-\,{\dfrac {5^{2}}{2^{8}}}\left\{\eta ^{8}-{\dfrac {1}{2^{4}\cdot 5}}{\dfrac {(R-c)^{2}R^{3}}{c'{}^{2}R'{}^{4}}}\right\}^{2}{\dfrac {(R-c)^{7}R^{8}}{c'{}^{7}R'{}^{12}}}\eta ^{8}=0.\end{matrix}}$

(60)

‎Таково уравнение, соответствующее октаэдрической форме $f(\eta _{1},\;\eta _{2})$.

Подставив в него значения постоянных:

$c=108,\ c'=1,$