Pagina:Werkecarlf03gausrich.djvu/363

$\gamma ^{\prime }\cos T+\gamma ^{\prime \prime }\sin T,$ sine respectu signi, minor quam $\gamma .$ Hinc concludimus, quoties $\varepsilon \gamma$ sit quantitas positiva, variabiles $E$ et $T$ semper simul crescere; quoties autem $\varepsilon \gamma$ sit quantitas negativa, necessario alteram variabilem semper decrescere, dum altera augeatur.

13.

Nexus inter variabiles $E$ et $T$ adhuc melius illustratur per ratiocinia sequentia. Statuendo ${\sqrt {\gamma \gamma -1}}=\delta ,$ ita ut fiat $\delta \delta =\alpha \alpha +\beta \beta =\gamma ^{\prime }\gamma ^{\prime }+\gamma ^{\prime \prime }\gamma ^{\prime \prime },$ ex aequationibus 20, 21, 22 deducimus

${\begin{aligned}&t(\delta +\alpha \cos E+\beta \sin E)\\&\quad =\gamma \delta +\alpha \alpha +\beta \beta +(\gamma ^{\prime }\delta +\alpha \alpha ^{\prime }+\beta \beta ^{\prime })\cos T+(\gamma ^{\prime \prime }\delta +\alpha \alpha ^{\prime \prime }+\beta \beta ^{\prime \prime })\sin T\\&\quad =(\gamma +\delta )(\delta +\gamma ^{\prime }\cos T+\gamma ^{\prime \prime }\sin T)\end{aligned}}$

Perinde ex aequationibus 21, 22 sequitur

$t(\alpha \sin E-\beta \cos E)=\varepsilon (\gamma ^{\prime }\sin T-\gamma ^{\prime \prime }\cos T)$

Hae aequationes, statuendo

${\frac {\alpha }{\delta }}=\cos L,\quad$ ${\frac {\beta }{\delta }}=\sin L,\quad$ ${\frac {\gamma ^{\prime }}{\delta }}=\cos M,\quad$ ${\frac {\gamma ^{\prime \prime }}{\delta }}=\sin M$

nanciscuntur formam sequentem:

${\begin{aligned}t(1+\cos(E-L))&=(\gamma +\delta )(1+\cos(T-M))\\t\sin(E-L)&=\varepsilon \sin(T-M)\end{aligned}}$

unde fit per divisionem, propter $(\gamma +\delta )(\gamma -\delta )=1,$

${\begin{aligned}&\operatorname {tang} {\tfrac {1}{2}}(E-L)=\varepsilon (\gamma -\delta )\operatorname {tang} {\tfrac {1}{2}}(T-M)\\&\operatorname {tang} {\tfrac {1}{2}}(T-M)=\varepsilon (\gamma +\delta )\operatorname {tang} {\tfrac {1}{2}}(E-L)\end{aligned}}$

Hinc non solum eadem conclusio derivatur, ad quam in fine art. praec. deducti sumus, sed insuper etiam patet, si valor ipsius ${E}$ crescat $360$ gradibus, valorem ipsius $T$ tantundem vel crescere vel diminui, prout $\varepsilon \gamma$ sit vel quantitas positiva vel negativa. Ceterum statuendo $\delta =\operatorname {tang} N,$ $\gamma =\sec N,$ manifesto erit

$\gamma -\delta =\operatorname {tang} (45^{\circ }-{\tfrac {1}{2}}N),\quad$$\gamma +\delta =\operatorname {tang} (45^{\circ }+{\tfrac {1}{2}}N)$