Pagina:Werkecarlf03gausrich.djvu/368

Et quum integrale definitum ad dextram perinde transformare liceat, manifesto integrale

$\int {\frac {({\cos T}^{2}-{\sin T}^{2})dT}{2\pi {\sqrt {mm{\cos T}^{2}+nn{\sin T}^{2}}}}}$

exprimetur per seriem infinitam citissime convergentem

$-{\frac {2(m^{\prime }m^{\prime }-n^{\prime }n^{\prime })+4(m^{\prime \prime }m^{\prime \prime }-n^{\prime \prime }n^{\prime \prime })+s(m^{\prime \prime \prime }m^{\prime \prime \prime }-n^{\prime \prime \prime }n^{\prime \prime \prime })+{\text{etc. }}}{(mm-nn)\mu }}=-{\frac {\nu }{\mu }}$

Calculus numericus commodissime per logarithmos perficitur, si statuimus

${\tfrac {1}{4}}{\sqrt {mm-nn}}=\lambda ,\quad$${\tfrac {1}{4}}{\sqrt {m^{\prime }m^{\prime }-n^{\prime }n^{\prime }}}=\lambda ^{\prime },\quad$${\tfrac {1}{4}}{\sqrt {m^{\prime \prime }m^{\prime \prime }-n^{\prime \prime }n^{\prime \prime }}}=\lambda ^{\prime \prime }$ etc.

unde erit

$\lambda ^{\prime }={\frac {\lambda \lambda }{m^{\prime }}},\quad$$\lambda ^{\prime \prime }={\frac {\lambda ^{\prime }\lambda ^{\prime }}{m^{\prime \prime }}},\quad$$\lambda ^{\prime \prime \prime }={\frac {\lambda ^{\prime \prime }\lambda ^{\prime \prime }}{m^{\prime \prime \prime }}}$ etc. atque

$\nu ={\frac {2\lambda ^{\prime }\lambda ^{\prime }+4\lambda ^{\prime \prime }\lambda ^{\prime \prime }+8\lambda ^{\prime \prime \prime }\lambda ^{\prime \prime \prime }+{\text{etc.}}}{\lambda \lambda }}$

18.

Per methodum hic explicatam etiam integralia indefinita (a valore variabilis $=0$ inchoantia) maxima concinnitate assignare licet. Scilicet, si $T^{\prime \prime }$ perinde per $m^{\prime },$ $n^{\prime },$ $T^{\prime }$ determinari supponitur, uti $T^{\prime }$ per $m,$ $n,$ $T,$ ac perinde rursus $T^{\prime \prime \prime }$ per $m^{\prime \prime },$ $n^{\prime \prime },$ $T^{\prime \prime }$ et sic porro, etiam pro quovis valore determinato ipsius $T,$ valores terminorum serie $T,$ $T^{\prime },$ $T^{\prime \prime },$ $T^{\prime \prime \prime }$ etc. ad limitem $\theta$ citissime convergent, eritque

${\begin{aligned}&\int {\frac {\operatorname {d} T}{\sqrt {mm{\cos T}^{2}+nn{\sin T}^{2}}}}={\frac {\theta }{\mu }}\\&\int {\frac {({\cos T}^{2}-{\sin T}^{2})\operatorname {d} T}{\sqrt {mm{\cos T}^{2}+nn{\sin T}^{2}}}}=-{\frac {\nu \theta }{\mu }}+{\frac {\lambda ^{\prime }\cos T\sin T^{\prime }+2\lambda ^{\prime \prime }\cos T^{\prime }\sin T^{\prime \prime }+4\lambda ^{\prime \prime \prime }\cos T^{\prime \prime }\sin T^{\prime \prime \prime }+{\text{etc. }}}{\lambda \lambda }}\end{aligned}}$

Sed haec obiter hic addigitavisse sufficiat, quum ad institutum nostrum non sint necessaria.

19.

Quodsi iam statuimus $m={\sqrt {G+G^{\prime }}},$ $n={\sqrt {G+G^{\prime \prime }}},$ valores quantitatum $P,$ $Q$ facile ad transscendentes $\mu ,$ $\nu$ reducentur. Quum enim $P,$ $Q\operatorname {sint}$ valores integralium