Originally Posted by

**HallsofIvy** Is that really "$\displaystyle 2\frac{s}{4}$"? That is, of course, just $\displaystyle \frac{s}{2}$ but I suspect that the "2" is not supposed to be there.

If it were just $\displaystyle x_{1}(s) = 2\arcsin(\frac{s}{4}) + \frac{s}{4}\sqrt{1-\frac{s^{2}}{16}} $ then $\displaystyle x_{1}'(x)= 2\frac{1/4}{\sqrt{1- \frac{s^2}{16}}}+ \frac{1}{4}\sqrt{1- \frac{s^2}{16}}+ \frac{s}{8}\frac{1}{\sqrt{1- s^2/8}}(-\frac{s}{8})$

Now, $\displaystyle \frac{1}{2}\frac{1}{\sqrt{1- \frac{s^2}{16}}}- \frac{s^2}{8}\frac{1}{1-\frac{s^2}{16}}$$\displaystyle = \frac{1}{2}\frac{1- \frac{s^2}{16}}{\sqrt{1- \frac{x^2}{16}}}$$\displaystyle = \frac{1}{2}\sqrt{1- \frac{s^2}{16}}$