I am reading the Feynman courses. It is written inside the book that a sphere of radius $a$ charged by a surface charge $\sigma = \sigma_0cos(\theta)$ should generate an electric field with a moment of $p = \frac{4\pi \sigma_0 a^3}{3}$ at the exterior of the sphere, and inside the sphere, the field should be constant and shall be $\vec{E}=-\frac{\sigma_0}{3\epsilon_0}\vec{z}$.
In the book, it is noted to use 2 spheres uniformly volumic charged (the spheres are charged in an opposite way). Inside the intersection of the two spheres, we got a zero charge: $\rho_{total} = \rho_1 + \rho_2 = \rho_+ + \rho_- = 0$ However, outside the intersection, I "guessed" that the electric charge is something like $\rho \cdot d\cdot cos(\theta)$ (When I say I guessed, I red it here : https://physics.stackexchange.com/a/489351/318644 ). With this result, I succeed to show the good result for the moment and the electric field.
However, I don't know how can I really demonstrate this result in a rigorous way... I am not even sure to understand it properly. So, if someone could explain me the "demonstration" that the two spheres uniformly charged in a volumic way are similar to a surfacic charged sphere of $\sigma = \sigma_0 cos(\theta)$