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For the theoretical assessment of the 2'P helium fine structure to become comparable to the precision measurements that have been made, it is necessary that the theory be calculated through order a~me . In particular, the second-order contribution from the Breit and mass-polarization operators must be evaluated to an accuracy of 1% or so. In this work, for each of five possible intermediate state symmetries the Dalgarno-Lewis method is used to obtain the first-order perturbed wave function, from which the second-order energy follows by integrat&on. Both the perturbed and unperturbed wave functions are expanded in Hylleraas-type series with a progressively larger number of terms, the second-order energies being computed at each stage; up to 455 terms are used for 'P intermediate states and up to 286 for 'P, D, 'D, and 'F. The sequence of second-order energy results for each symmetry is extrapolated to the limit of an infinite number of basis functions to arrive at a final result. The P, 'P, and 'D states will contribute to both the larger and the smaller fine-structure intervals vo, and v», respectively, while 'I' and 'D states affect only v». The total theoretical result, up to order a mc, for vo, is much more accurate than that for v», allowing the finestructure constant 0, to be determined very precisely by comparison of theory to experiment, with the result a ' = 137.036 08(13), good to 0.94 ppm,

A calculation of higher-order relativistic contributions to the Zeeman effect in the 2'P, 2'S, 3'P, 4 P, and 5'P states of helium is given. The contributions to the g factors gs, g~, and g~, have been calculated to order a' and mlM where m is the electron mass and M is the helium mass.Relativistic contributions are obtained from the matrix elements of a reduced Breit HamHtonian with a series of numerical wave functions using a Hylleraas basis. The comparison of the theoretical values with the experhnental determinations of the g factors is given for the 2'P and 2'S states. The value of gs calculated for 2 P helium is within two standard deviations of the experimental error of the measurement of Lewis et al. A recent precise measurement of the g factor in 2'8, helium by Brossel et aL is three standard deviations of the experimental error from the value calculated here and from the result of Perl and Hughes. These higher-order contributions are necessary for the determination of the 2'P fine-structure intervals from the Zeeman effect, and hence to test the theory of fine structure for helium and to d~aine precisely the fine-structure constant a from the helium fine structure. The relativistic contribuuusss to gs have been calculated for the helium isoelectronic series from Li ii through Ne ix for the 2 'P "3 'P "4 'P "and 5 'P, states.

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