To derive the angles and expressions for the spherical triangle $ABC$ using the coordinates $\psi, \theta, \psi', \theta'$ and $x$, we follow the setup given in the figure and the explanation:

1. Coordinate System Setup:

   • The $z$-axis points towards the vertex $A$.
   • The $z'$-axis points towards vertex $B$.
   • The vertex $C$ corresponds to the point $P$.

2. Expressions for the angles:

   • $\psi$ (the angle between the $x$-axis and the line connecting $O$ and $C$ in the $xy$-plane):

     • Given by $\psi = A - 90^\circ$, where $A$ is the angle at vertex $A$.

   • $\theta$ (the angle between the $z$-axis and the line connecting $O$ and $C$ in the $yz$-plane):

     • Given by $\theta = 90^\circ - b$, where $b$ is the side opposite vertex $B$.

   • $\psi'$ (the angle between the $x'$-axis and the line connecting $O$ and $C$ in the $x'y'$-plane):

     • Given by $\psi' = 90^\circ - B$, where $B$ is the angle at vertex $B$.

   • $\theta'$ (the angle between the $z'$-axis and the line connecting $O$ and $C$ in the $y'z'$-plane):

     • Given by $\theta' = 90^\circ - a$, where $a$ is the side opposite vertex $A$.

   • $x$ (the distance between $A$ and $B$ along the sphere):

     • Given by $x = c$, where $c$ is the side opposite vertex $C$.

Thus, the derived expressions for the angles and the sides are: $\psi = A - 90^\circ, \quad \theta = 90^\circ - b, \quad \psi' = 90^\circ - B, \quad \theta' = 90^\circ - a, \quad x = c.$

Would you like more details on any specific step or have any questions?

Here are some questions you might want to ask next:

1. Can you explain the derivation of these expressions in more detail?
2. How do these coordinate transformations help in solving spherical triangles?
3. What are some applications of spherical trigonometry?
4. Can you show how to solve a specific problem involving spherical triangles?
5. How do these angles relate to each other in a spherical triangle?

Tip: When dealing with spherical triangles, always remember that the sum of the angles exceeds 180 degrees, unlike in planar triangles.