We carry out a phase-shift analysis of $p-p$ and $n-p$ elastic scattering data in the laboratory kinetic energy range of 47.5 to 60.9 MeV. Despite the inclusion of new $n-p$ total and differential cross-section data since the phase-shift analyses of MacGregor, Arndt, and Wright (papers VI and X in particular), the ${\chi }^2$ vs ${\epsilon }_1$ curve retains its double minimum, and the anomalous value for the phase parameter $\delta \left({}_{}{}^1{}_{}{}^{}P_1^{}{}_{}{}^{}\right)$ persists. The data yield a range of solutions rather than a unique $I=0\mathrm{}$ phase-shift solution, and, in fact, the ${\chi }^2$ vs ${\epsilon }_1$ curve is even flatter than it was in paper VI. The allowed range of ${\epsilon }_1$ is found to be from -10° to +3°, approximately. To find a unique solution, we constrain ${\epsilon }_1$ to have a reasonable theoretical value of +2.78°, and present the corresponding constrained phase-shift solution. This yields $\delta \left({}_{}{}^1{}_{}{}^{}P_1^{}{}_{}{}^{}\right)=-3.52\mathrm{}°\pm 1.04°$, which is 4.5 standard deviations or more above predictions by meson-theoretical models. In fact, throughout the allowed range of ${\epsilon }_1$, the searched value of $\delta \left({}_{}{}^1{}_{}{}^{}P_1^{}{}_{}{}^{}\right)$ remains at least this far above theoretical predictions. We determine that the Harwell $n-p$ $\frac{d\sigma }{d\Omega }$ data at extreme forward and extreme backward angles are responsible for the high value of $\delta \left({}_{}{}^1{}_{}{}^{}P_1^{}{}_{}{}^{}\right)$, and recommend that these measurements be retaken. We emphasize that contrary to popular belief, it is not sufficient just to fix the relative backward $n-p$ $\frac{d\sigma }{d\Omega }$, because incorrect forward values will result in a wrong value for $\delta \left({}_{}{}^1{}_{}{}^{}P_1^{}{}_{}{}^{}\right)$. With regard to forward data, good extreme forward absolute $\frac{d\sigma }{d\Omega }$ data will be more effective than relative forward data spanning the 0°-90° range. Finally, with respect to ${\epsilon }_1$, we emphasize that the existing types of $n-p$ data ($\frac{d\sigma }{d\Omega }$, ${\sigma }_{\mathrm{tot}}$, and $P$) will not remove the ambiguity in this phase parameter. Some other type of experiment must be done, as we intend to discuss in a succeeding paper.

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