According to the idea of spontaneous breakdown of the symmetry, we study mass relations among the distinct multiplets of a group. We first consider an interacting system which possesses the symmetry SU(3) (unimodular unitary group). Introducing the mass counter terms with the broken symmetry, we seek for self-consistent solutions to the vertex function of the gravitational field (i.e. masses of the particles). It is assumed, in this paper, that the mass splitting among particles which belong to a common unitary-multiplet should satisfy the Gell-Mann-Okubo mass formula. Ratios among the parameters which are contained in the Gell-Mann-Okubo formula (asymmetry parameters of the meson and baryon masses) are determined by the existence condition of the asymmetry solutions to our mass equation. It is a natural consequence from the spontaneous breakdown that the characteristic magnitude of the mass splitting for the pseudoscalar mesons (π, η K and K) is about (m/µ) times as large as that for the baryons (Σ, Λ, N and Ξ). Corret signs of the mass asymmetry parameters are also predicted by the choice of the F-D mixing ratio ξ≡F/D ∼0.7. The magnitude of the meson mass splitting is a little large in our approximation.