### The shortest distance between cities

The distance $l=R\gamma$, where $\gamma$ is the angle between the vectors $\vec{OP_1}$ and $\vec{OP_2}$ ( $O$ is the center). One has
$\vec{OP_1}=R(\cos\alpha_1\cos\beta_1, \cos\alpha_1 \sin\beta_1,\sin\alpha_1)\\ \vec{OP_2}=R(\cos\alpha_2\cos\beta_2, \cos\alpha_2 \sin\beta_2,\sin\alpha_2)\\\cos\gamma=\vec{OP_1}\cdot\vec{OP_2}/|\vec{OP_1}|\cdot|\vec{OP_2}|=\cos\alpha_1\cos\alpha_2\cos(\beta_1-\beta_2)+\sin\alpha_1+\sin\alpha_2$