Relativistic partial-wave dispersion relations are formulated for elastic nucleon-nucleon scattering. These dispersion relations are integral equations with an inhomogeneous term taken from single-particle exchange contributions. The particles under consideration are $\ensuremath{\pi}(I=1, \mathrm{pseudoscalar})$, $\ensuremath{\eta}(I=0, \mathrm{pseudoscalar})$, $\ensuremath{\rho}(I=1,\mathrm{vector})$, $\ensuremath{\omega}(I=0, \mathrm{vector})$, $\ensuremath{\phi}(I=0, \mathrm{vector})$, and $\ensuremath{\sigma}(I=0, \mathrm{scalar})$. The existence of a $\ensuremath{\sigma}$ meson is not well established. Two possibilities are considered: (i) The $\ensuremath{\sigma}$ meson exists, in which case the mass and coupling constants are taken to be two parameters of the present problem. (ii) The $\ensuremath{\sigma}$ meson does not exist but the $I=0$, $J=0$ two-pion continuum is taken into account. This two-pion continuum can be treated as a superposition of scalar particles with a mass spectrum determined by pion-nucleon and pion-pion interactions. Information on the $\ensuremath{\pi}N$ interaction is obtained from $\ensuremath{\pi}N$ scattering data, while the $S$-wave $\ensuremath{\pi}\ensuremath{\pi}$ interaction is represented with a relativistic scattering-length approximation. In addition to the $\ensuremath{\pi}\ensuremath{\pi}$ scattering length, a cutoff on the two-pion spectrum is introduced. Thus two parameters are introduced in either (i) or (ii). Aside from the masses and coupling constants of the particles mentioned, a cutoff parameter is needed for each of the vector mesons $\ensuremath{\rho}$, $\ensuremath{\omega}$, and $\ensuremath{\phi}$. These are taken to be coefficients in an exponentially decreasing factor suggested by the Regge-pole behavior of composite particles. A total of twelve adjustable parameters is used and a search program is formulated to fit 560 $\mathrm{pp}$ and $\mathrm{np}$ data collected by the Livermore group ranging from 9.68 to 388 MeV. In both cases (i) and (ii), a fit is obtained with a "goodness to fit" value of approximately 8%, meaning that the ${\ensuremath{\chi}}^{2}$ is \ensuremath{\sim}548 if the uncertainty inherent in the theory is assumed to be 8%.