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The variational method, within the Hamiltonian formalism of reformulated QEDis used to determine relativistic wave equations for a system of three fermionsof arbitrary mass interacting electromagnetically. The interaction kernels ofthe equations are, in essence, the invariant $\mathcal{M}$ matrices in lowestorder. The equations are used to obtain relativistic $O(\alpha^2)$ correctionsto the non-relativistic ground state energy levels of the Muonium negative ion($\mu^+ e^- e^-$) as well as of $\mathrm{Ps}^-$ and $\mathrm{H}^-$, usingapproximate variational three-body wave functions. The results are comparedwith other calculations, where available. The relativistic correction forMu$^-$ is found to be $-1.0773 \times 10^{-4}$ eV.

Relativistic Three-Fermion Wave Equations in Reformulated QED and Relativistic Effects in Muonium Minus