## Abstract

A liquid-hydrogen-cooled discharge tube similar to that formerly used in the investigation of the fine structure of the $\alpha $-line of heavy hydrogen has now been used to reduce the Doppler width of the components of the line n = 4$\rightarrow $n = 3 in the spectrum of He$^{+}$. The resolving power necessary to separate some components was obtained without sacrificing spectral range by using two Fabry-Perot etalons in series mounted externally to a prism spectrograph. For other components a single etalon was used. Eight components were resolved and their separations measured, and a ninth partially resolved. The positions of certain band lines of hydrogen and helium which lie close to the He$^{+}$ complex and overlap it in different orders of interference were also measured. The relative intensities of the components were broadly in agreement with the relative transition probabilities calculated from the Dirac theory, but some anomalies were observed. From the measurements it is deduced that some of the energy levels are displaced from the positions predicted by the Dirac theory: the 3S$_{\frac{1}{2}}$ level upwards by 0$\cdot $140 $\pm $ 0$\cdot $005 cm$^{-1}$, the 4S$_{\frac{1}{2}}$ level upwards by 0$\cdot $056 $\pm $ 0$\cdot $003 cm$^{-1}$ and the 4P$_{\frac{1}{2}}$ level downwards by 0$\cdot $011 $\pm $ 0$\cdot $003 cm$^{-1}$. These displacements of the S$_{\frac{1}{2}}$ levels are in good agreement with the predictions of the new theory of quantum electrodynamics, but the theory does not predict a displacement of the 4P$_{\frac{1}{2}}$ level by this amount. The Stark effect could account for part of the discrepancy, but there remains a disagreement with the theory which exceeds the estimated experimental error.