We recently proved that " the center of buoyancy is equal to the center of hydrostatic pressure " for floating bodies. This subject was an unsolved problem in physics and naval architecture, even though the buoyancy taught by Archimedes' principle can be obtained clearly by the surface integral of hydrostatic pressure. Then we thought that the reason why the vertical position of the center of pressure could not be determined was that the horizontal force would be zero due to equilibrium in the upright state.

As a breakthrough, we dared to create the left-right asymmetric pressure field by inclining the floating body with heel angle. In that state, the forces and moments due to hydrostatic pressure were calculated correctly with respect to the tilted coordinate system fixed to the body. By doing so, we succeeded in determining the center of pressure. Then, by setting the heel angle to zero in order to make it upright state, it could be proved that the center of hydrostatic pressure is equal to the well-known center of buoyancy, i.e., the centroid of the cross-sectional area under the water surface.

As mentioned above, we have already proved this problem for rectangular and arbitrarily shaped cross-sections, and published them on this viXra.org in English. Following that, in the 2nd report, separate proofs for a semi-submerged circular cylinder and triangular prism were also published here. Thus, we have completed the proof for floating bodies, so in this 3rd report, we aim to prove for submerged bodies. We first prove for a submerged circular cylinder, and then apply Gauss's integral theorem to prove it clearly for an arbitrarily shaped submerged body.