Order ${\alpha }^2$ Electrodynamic Corrections to the Lamb Shift

The fourth-order radiative correction to the slope at $q^2=0\mathrm{}$ of the Dirac form factor of the free-electron vertex is calculated using computer techniques. The result, $m^2\frac{\partial {F_1}^{\mathrm{}\left(4\right)\mathrm{}}\mathrm{}\left(0\right)\mathrm{}}{\partial q^2}={\left(\frac{\alpha }{\pi }\right)}^2[0.48\mathrm{}\pm 0.07],$ disagrees with previous calculations, and implies a new theoretical value for the order ${\alpha }^2{\left(Z\alpha \right)}^{4}m{c}^2$ contribution to the Lamb shift. The new values for the $2S_{\frac{1}{2}}-2\mathrm{}{P}_{\frac{1}{2}}$ separation in H and D are increased by (0.35 ± 0.07) MHz and are in good agreement with the results of recent experiments.