We examine the consequences of the hypothesis that, for some range of energies, the second-order terms in the perturbation expansion of the Fermi theory are small but non-negligible corrections to the first-order terms, and responsible for reactions which violate first-order selection rules. We compute matrix elements through second order in the weak coupling constant $G$ and introduce subtraction constants which are necessary to render the matrix elements finite to this order. From this point of view we consider a number of weak-interaction problems, including the experimental parameters of $\mu$ decay, the universality of the $\mu$-decay coupling constant and the Fermi $\beta$-decay coupling constant, $K_{l3\mathrm{}}$ decays, and $\Delta S=\Delta Q$ and $\Delta S=2\mathrm{}$ semileptonic decays.

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