= Solution
Use <arc length> $s$ as the <affine parameter>, and put a dot over derivatives with respect to $s$. The <geodesic Lagrangian> on the <sphere> is
$$
L=\frac{a^2}{2}(\dot\theta^2+\sin^2\theta\,\dot\phi^2).
$$
The <cyclic coordinate> $\phi$ gives $\partial L/\partial\dot\phi=a^2\sin^2\theta\,\dot\phi=\text{constant}$. With the constant normalized as in the paper,
$$
\boxed{\sin^2\theta\,\dot\phi=h.}
$$
Here $h$ has units of inverse length, because $s$ is physical <arc length>. Unit-speed normalization gives
$$
a^2\dot\theta^2+\frac{a^2h^2}{\sin^2\theta}=1.
$$
For $h\ne0$, division by $\dot\phi^2$ yields the <spherical-geodesic first integral>
$$
\boxed{\left(\frac{d\theta}{d\phi}\right)^2+V_1(\theta)=0,
\qquad V_1(\theta)=\sin^2\theta-\frac{\sin^4\theta}{a^2h^2}.}
$$
The turning latitudes satisfy $\sin\theta=|ah|$. In the usual polar-coordinate range $0\le\theta\le\pi$,
$$
\boxed{\theta_{\min}=\arcsin|ah|,\qquad
\theta_{\max}=\pi-\arcsin|ah|,\qquad 0<|ah|\le1.}
$$
The maximum departure from the equator, often the geometric quantity intended by an angular amplitude, is instead
$$
\boxed{\max|\theta-\pi/2|=\arccos|ah|.}
$$
Thus \b[<polar angle> and signed latitude must not be interchanged]. At $|ah|=1$ the <geodesic> is the equator. At $h=0$ it is a meridian through the poles; $\phi$ cannot serve as a global parameter and the divided equation is inapplicable. The limiting polar range is then $[0,\pi]$.
The original PDF prints an ambiguous derivative with the square on $\theta$ rather than a conventional derivative notation. If the intended equation was second order, differentiating the first integral, or eliminating $\dot\theta$ from the <Euler-Lagrange equation>, gives the equally useful form
$$
\boxed{\frac{d^2\theta}{d\phi^2}+V_2(\theta)=0,\qquad
V_2(\theta)=\sin\theta\cos\theta-
\frac{2\sin^3\theta\cos\theta}{a^2h^2}.}
$$
It has the same normalized <geodesics> and the same turning angles. Giving both forms resolves the typographical ambiguity without pretending that a second derivative is the square of a first derivative. Geometrically these trajectories are <great circles>, with inclination $\arccos|ah|$ to the equator.
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