Nonrelativistic eigenvalues accurate to one part in ${10}^9$ to ${10}^{10}$ have been obtained for the $2{}_{}{}^3{}_{}{}^{}P$ and $3{}_{}{}^3{}_{}{}^{}P$ states of helium by improving the method previously developed for calculating two-electron atom $P$-state wave functions, and by including up to 560 terms in the expansion. The corresponding values for the fine-structure splitting of the two levels are accurate to between ${10}^{-5\mathrm{}}$ and ${10}^{-6\mathrm{}}$ ${\mathrm{cm}}^{-1\mathrm{}}$, and the small discrepancy with experiment is presumed to be due to the neglect of quantum-electrodynamic correction terms of relative order ${\alpha }^2$ in the calculation. The results for the $2{}_{}{}^3{}_{}{}^{}P$ state are in good agreement with those obtained recently by C. Schwartz.

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