in which G = exp ⁡ ⁡ G , β β i = − − γ γ i ⋅ ⋅ P ^ ^ , γ γ i ⊥ ⊥ μ μ = ( η η μ μ ν ν + P ^ ^ μ μ P ^ ^ ν ν ) γ γ ν ν i , Σ Σ i = γ γ 5 i β β i γ γ ⊥ ⊥ i , P i ≡ ≡ p ⊥ ⊥ − − i 2 Σ Σ i ⋅ ⋅ ∂ ∂ G Σ Σ i , i = 1 , 2. The (covariant) structure of these equations are analogous to those of a Dirac equation for each of the two particles, with M i and E i playing the roles that m + S and ε ε − − A do in the single particle Dirac equation ( γ γ ⋅ ⋅ p − − β β ( ε ε − − A ) + m + S ) ψ ψ = 0. Over and above the usual kinetic part γ γ 1 ⋅ ⋅ p ⊥ ⊥ and time-like vector and scalar potential portions, the spin-dependent modifications involving Σ Σ i ⋅ ⋅ ∂ ∂ G Σ Σ i and the last set of derivative terms are two-body recoil effects absent for the one-body Dirac equation but essential for the compatibility (consistency) of the two-body equations. The connections between what are designated as the vertex invariants L , G and the mass and energy potentials M i , E i are M 1 = m 1 cosh ⁡ ⁡ L + m 2 sinh ⁡ ⁡ L , M 2 = m 2 cosh ⁡ ⁡ L + m 1 sinh ⁡ ⁡ L , E 1 = ε ε 1 cosh ⁡ ⁡ G − − ε ε 2 sinh ⁡ ⁡ G , E 2 = ε ε 2 cosh ⁡ ⁡ G − − ε ε 1 sinh ⁡ ⁡ G . Comparing Eq.(16) with the first equation of this article one finds that the spin-dependent vector interactions are A ~ ~ 1 μ μ = ( ( ε ε 1 − − E 1 ) ) P ^ ^ μ μ + ( 1 − − G ) p ⊥ ⊥ μ μ − − i 2 ∂ ∂ G ⋅ ⋅ γ γ 2 γ γ 2 μ μ , A 2 μ μ = ( ( ε ε 2 − − E 2 ) ) P ^ ^ μ μ − − ( 1 − − G ) p ⊥ ⊥ μ μ + i 2 ∂ ∂ G ⋅ ⋅ γ γ 1 γ γ 1 μ μ , Note that the first portion of the vector potentials is timelike (parallel to P ^ ^ μ μ ) while the next portion is spacelike (perpendicular to P ^ ^ μ μ ) . The spin-dependent scalar potentials S ~ ~ i are S ~ ~ 1 = M 1 − − m 1 − − i 2 G γ γ 2 ⋅ ⋅ ∂ ∂ L , S ~ ~ 2 = M 2 − − m 2 + i 2 G γ γ 1 ⋅ ⋅ ∂ ∂ L .