From the geographical latitudes of two places $\left(A: 47^{\circ} 6^{\prime}, B: 59^{\circ} 56^{\prime} 30^{\prime \prime}\right)$ and their time difference ( $\left.59^{\prime} 29^{\prime \prime}\right)$, determine their distance from each other, $(r=6378,3 \mathrm{~km})$.
Let $\varphi_{1}$ and $\varphi_{2}$ be the latitudes of the two places. The distance $D$ between the two places is calculated from a spherical triangle, one side of which is $90^{\circ}-\varphi_{1}$, the other side is $90^{\circ}-\varphi_{2}$, and the angle between them is the difference in longitudes $\lambda$ of the two places, which is equal to 15 times the time difference. Then, as is known,

$$
\cos D=\frac{\sin \varphi_{1} \sin \left(\varphi_{2}+\psi\right)}{\cos \psi}
$$

if

$$
\operatorname{tg} \psi=\operatorname{ctg} \varphi_{1} \cos \lambda
$$

Substituting the given values, $D=15^{\circ} 31^{\prime}$. Since

$$
2 r \pi: x=360^{\circ}: D
$$

the sought distance in kilometers is:

$1727 \mathrm{~km}$.

(Paulin Rássy, Eger.)

The problem was also solved by: Bartók I., Enyedi B., Haar A., Liebner A., Losonczy I., Riesz K., Róth A., Szűcs A.