aus Wikisource, der freien Quellensammlung
Vierter Abschnitt. §. 49.
Aus den Gleichungen (13) und (14) leiten wir ab:
1
η
φ
1
(
1
η
)
=
∑
0
∞
1
p
k
+
q
k
η
,
1
ζ
χ
1
(
1
ζ
)
=
∑
0
∞
1
(
p
k
c
a
+
q
k
)
ζ
−
b
a
p
k
=
∑
0
∞
1
(
p
k
c
a
+
q
k
)
(
c
b
−
a
b
η
)
−
b
a
p
k
.
{\displaystyle {\begin{aligned}{\frac {1}{\eta }}\,\varphi _{1}\left({\frac {1}{\eta }}\right)&=\sum _{0}^{\infty }\,{\frac {1}{p_{k}+q_{k}\,\eta }},\\\,\\{\frac {1}{\zeta }}\,\chi _{1}\left({\frac {1}{\zeta }}\right)&=\sum _{0}^{\infty }\,{\frac {1}{\left(p_{k}{\frac {c}{a}}+q_{k}\right)\zeta -\,{\frac {b}{a}}\,p_{k}}}\\&=\sum _{0}^{\infty }\,{\frac {1}{\left(p_{k}{\frac {c}{a}}+q_{k}\right)\left({\frac {c}{b}}-{\frac {a}{b}}\,\eta \right)-\,{\frac {b}{a}}\,p_{k}}}.\\\end{aligned}}}
Wir disponiren nun über die Coefficienten so, dass das
k
{\displaystyle k\!}
1
ζ
χ
1
(
1
ζ
)
{\displaystyle {\frac {1}{\zeta }}\,\chi _{1}\left({\frac {1}{\zeta }}\right)}
(
k
+
1
)
{\displaystyle (k+1)\!}
1
η
φ
1
(
1
η
)
{\displaystyle {\frac {1}{\eta }}\,\varphi _{1}\left({\frac {1}{\eta }}\right)}
(15)
1
η
φ
1
(
1
η
)
−
1
ζ
χ
1
(
1
ζ
)
=
1
p
0
+
q
0
η
.
{\displaystyle {\frac {1}{\eta }}\,\varphi _{1}\left({\frac {1}{\eta }}\right)-\,{\frac {1}{\zeta }}\,\chi _{1}\left({\frac {1}{\zeta }}\right)=\,{\frac {1}{p_{0}+q_{0}\,\eta }}.\,}
Damit dies zu Stande komme, hat man
(16)
p
k
+
1
=
p
k
c
2
−
b
2
a
b
+
q
k
c
b
q
k
+
1
=
−
1
b
(
p
k
c
+
q
k
a
)
{\displaystyle {\begin{aligned}&p_{k+1}=p_{k}\,{\frac {c^{2}-b^{2}}{ab}}+q_{k}\,{\frac {c}{b}}\\\,\\&q_{k+1}=-\,{\frac {1}{b}}\,(p_{k}c+q_{k}a)\\\end{aligned}}}
k
{\displaystyle k\!}
p
k
=
P
α
k
,
q
k
=
Q
α
k
.
{\displaystyle {\begin{aligned}&p_{k}={\mathfrak {P}}\alpha ^{k},\\\,\\&q_{k}={\mathfrak {Q}}\alpha ^{k}.\\\end{aligned}}}
(17)
P
(
α
−
c
2
−
b
2
a
b
)
=
Q
c
b
,
Q
(
α
+
a
b
)
=
−
P
c
b
,
{\displaystyle {\begin{aligned}&{\mathfrak {P}}\left(\alpha -\,{\frac {c^{2}-b^{2}}{ab}}\right)={\mathfrak {Q}}\,{\frac {c}{b}},\,\\\,\\&{\mathfrak {Q}}\left(\alpha +\,{\frac {a}{b}}\right)=-{\mathfrak {P}}\,{\frac {c}{b}},\\\end{aligned}}}
P
{\displaystyle {\mathfrak {P}}\!}
Q
{\displaystyle {\mathfrak {Q}}\!}
α
{\displaystyle \alpha \!}
(18)
α
2
+
α
a
2
+
b
2
−
c
2
a
b
+
1
=
0.
{\displaystyle \alpha ^{2}+\alpha \,{\frac {a^{2}+b^{2}-c^{2}}{ab}}+1=0.\,}
α
1
{\displaystyle \alpha _{1}\!}
α
2
{\displaystyle \alpha _{2}\!}
