4.22. Let's introduce a coordinate system with the origin $O$ at the center of the Earth, with the axes $O x$ and $O y$ passing through the points on the equator with longitudes $0^{\circ}$ and $90^{\circ}$ respectively, and the axis $\mathrm{Oz}$ passing through the North Pole. What are the coordinates of a point on the Earth's surface with latitude $\varphi$ and longitude $\psi$? (Consider the Earth as a sphere of radius $R$; in the southern hemisphere, the latitude is taken as negative.)
4.22. Let $p=(x, y, z)$ be a given point on the Earth's surface, $P^{\prime}$ - the projection on the equatorial plane. Then $z=R \sin \varphi$ and $O P^{\prime}=R \cos \varphi$. Therefore, $x=O P^{\prime} \cos \psi=$ $=R \cos \varphi \cos \psi$ and $y=R \cos \varphi \sin \psi$. Thus, $p=(R \cos \varphi \cos \psi$, $R \cos \varphi \sin \psi, R \sin \varphi)$.