The Dirac equation for the electron is solved to terms that are second order in the applied field by the application of the Foldy-Wouthuysen transformation. In the process this generates all the terms that are necessary to ensure that thegtensor is gauge-invariant to the third order. An explicit expression is given for the third-ordergtensor and the result is compared with the formula given by Tippins [1]. A term that reflects the velocity dependence by the Bohr magneton gives a significant contribution especially for atoms of high atomic number. This same term is additional to the extra term obtained by Stone [2] and is of the same magnitude: it must be invoked to account for thegfactor of atomic hydrogen.