Show that the tractrix discussed in Example 1.17 is orthogonal to the lower half of each circle with radius $a$ and center on the positive $y$-axis.
Example 1.17:
$$\left(y^{\prime}\right)^2=x^2y^{\prime\prime}$$
We note that $y$ is missing, so we make the substitutions $p=y^{\prime}$ and $p^{\prime}=y^{\prime\prime}$.
Thus the equation becomes:
$$p^2=x^2p^{\prime}$$
Now use separation of variables:
$$\frac{dx}{x^2}=\frac{dp}{p^2}$$
This integrates to:
$$-\frac{1}{x}=-\frac{1}{p}+E$$
for some unknown $E$. We resubstitute $p=y^{\prime}$ and integrate to obtain:
$$y(x)=\frac{x}{E}-\frac{1}{E^2} \ln(1+Ex)+D$$
