Precision measurements of the energy differences ${\mathcal{S}}_4(4\mathrm{}{}_{}{}^2{}_{}{}^{}S_{\frac{1}{2}}^{}{}_{}{}^{}-4\mathrm{}{}_{}{}^2{}_{}{}^{}P_{\frac{1}{2}}^{}{}_{}{}^{})$ and $\Delta E_4-{\mathcal{S}}_4(4\mathrm{}{}_{}{}^2{}_{}{}^{}P_{\frac{3}{2}}^{}{}_{}{}^{}-4\mathrm{}{}_{}{}^2{}_{}{}^{}S_{\frac{1}{2}}^{}{}_{}{}^{})$ in ${\left({\mathrm{He}}^4\right)}^{+}$ are reported. The experimental results, ${\mathcal{S}}_4=1768\mathrm{}\pm 5$ MHz and $\Delta E_4-{\mathcal{S}}_4=20179.7\mathrm{}\pm 1.2$ MHz, agree with the values predicted by quantum electrodynamic theory, ${\mathcal{S}}_4=1768.34\mathrm{}\pm 0.51$ MHz and $\Delta E_4-{\mathcal{S}}_4=20180.78\mathrm{}\pm 0.56$ MHz. An electron gun excites the states of interest in a section of waveguide containing helium gas. A magnetic field applied perpendicular to the waveguide axis is used to scan a microwave resonance between suitable Zeeman levels for a fixed frequency of oscillating electric field in the waveguide. Any induced electric dipole transitions $4{}_{}{}^2{}_{}{}^{}S_{\frac{1}{2}}^{}{}_{}{}^{}\to 4{}_{}{}^2{}_{}{}^{}P_{\frac{1}{2}}^{}{}_{}{}^{}$ or $4{}_{}{}^2{}_{}{}^{}S_{\frac{1}{2}}^{}{}_{}{}^{}\to 4{}_{}{}^2{}_{}{}^{}P_{\frac{3}{2}}^{}{}_{}{}^{}$ reduce the intensity of 1215-Å radiation which is emitted in the natural decay of $4S$ to $2P$. This light is directed onto an ultraviolet-detecting phototube whose output is measured by a lock-in detector for which the synchronous signal is provided by square-wave amplitude modulation of the microwave field. The resonances obtained by varying the magnetic field are fitted to a theoretical line-shape formula by a computerized least-squares program. The resulting best-fit parameters include either ${\mathcal{S}}_4$ or $\Delta E_4-{\mathcal{S}}_4$, depending on the transition studied. Consideration is given to the dependence of the resonance center on the gun current, helium pressure, and microwave power level.

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