Superconductivity and magnetism in the doping series U$_{2}$Rh$_{\mathrm{x}}$Pt$_{\mathrm{(1-x)}}$C$_{2}$

U$_{2}$PtC$_{2}$ has been known for many years to exhibit nearly-heavy-fermion behavior, as well as superconductivity, T$_{\mathrm{c}}$ $\sim$ 1.5 K. Little is known about the nature of the superconductivity, but many other uranium based heavy fermion superconductors, such as UPt$_{3}$ and UBe$_{13}$, have been shown to be unconventional. U$_{2}$RhC$_{2}$ also shows nearly-heavy-fermion behavior, but it is non-superconducting and reported to be antiferromagnetic, T$_{\mathrm{N}}$ $\sim$ 18 K. These observations have motivated our study of the doping series of U$_{2}$Rh$_{\mathrm{x}}$Pt$_{\mathrm{(1-x)}}$C$_{2}$ in order to investigate the evolution from the antiferromagnetic to the superconducting groundstate, as well as the role of the antiferromagnetism in the superconductivity. Through measurement of the resistivity, magnetic susceptibility and heat capacity of polycrystalline samples, we show the suppression of antiferromagnetism, the presence of competing ferromagnetism, and emergence of superconductivity with doping. Furthermore, we present evidence that the emergence of superconductivity, which deviates from single-gap BCS theory, is not directly related to the suppression of antiferromagnetism.