Article 23
Derivatives of Gudermanian and Inverse.

Let

Prob. 61.

Differentiate:

$$\begin{array}{llllllll}\hfill y& =sinhu-gdu,& \hfill y& =sinv+{gd}^{-1}v,& \hfill & & \hfill & \\ \hfill y& =tanhusechu+gdu,& \hfill y& =tanvsecv+{gd}^{-1}v.& \hfill & & \hfill & \\ \hfill & & \hfill & & \hfill \end{array}$$

Prob. 62.

Writing the “elliptic integral of the first kind” in the form

$$u={\int \; <!--nolimits-->}_0^{\phi }\frac{d\phi }{\sqrt{1-{\kappa }^2{sin}^2\phi }},$$

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

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

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

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