  The formalism relating the relativistic three-particle infinite-volume
scattering amplitude to the finite-volume spectrum has been developed thus far
only for identical or degenerate particles. We provide the generalization to
the case of three nondegenerate scalar particles with arbitrary masses. A key
quantity in this formalism is the quantization condition, which relates the
spectrum to an intermediate K matrix. We derive three versions of this
quantization condition, each a natural generalization of the corresponding
results for identical particles. In each case we also determine the integral
equations relating the intermediate K matrix to the three-particle scattering
amplitude, $\mathcal M_3$. The version that is likely to be most practical
involves a single Lorentz-invariant intermediate K matrix, $\widetilde{\mathcal
K}_{\rm df,3}$. The other versions involve a matrix of K matrices, with
elements distinguished by the choice of which initial and final particles are
the spectators. Our approach should allow a straightforward generalization of
the relativistic approach to all other three-particle systems of interest.
