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## Article 23Derivatives of Gudermanian and Inverse.

Let

Prob. 61.
Diﬀerentiate:

$\begin{array}{llllllll}\hfill y& =sinhu-gdu,\phantom{\rule{2em}{0ex}}& \hfill y& =sinv+{gd}^{-1}v,\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill y& =tanhusechu+gdu,\phantom{\rule{2em}{0ex}}& \hfill y& =tanvsecv+{gd}^{-1}v.\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill \end{array}$
Prob. 62.
Writing the “elliptic integral of the ﬁrst kind” in the form
 $u={\int }_{0}^{\phi }\frac{d\phi }{\sqrt{1-{\kappa }^{2}{sin}^{2}\phi }},$

$\kappa$ being called the modulus, and $\phi$ the amplitude; that is,

show that, in the special case when $\kappa =1$,

$\begin{array}{llllllllllll}\hfill u& ={gd}^{-1}\phi ,\phantom{\rule{2em}{0ex}}& \hfill amu& =gdu,\phantom{\rule{2em}{0ex}}& \hfill sinamu& =tanhu,\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill cosamu& =sechu,\phantom{\rule{2em}{0ex}}& \hfill tanamu& =sinhu;\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

and that thus the elliptic functions $sinamu$, etc., degenerate into the hyperbolic functions, when the modulus is unity.3

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