Theoretical formulas that include finite-nuclear-size corrections and higher-order matrix elements are given for the analysis of first-forbidden beta transitions. The formulas are arranged so that the contribution of the higher-order matrix elements can be seen clearly. Examples are given that illustrate when the higher-order terms must be included and when they can be neglected. The procedure for determining nuclear-matrix elements with a computer is discussed. Particular attention is given to the problem of setting limits of error. Nuclear-matrix elements for ${\mathrm{Rb}}^{86}$ and ${\mathrm{Rb}}^{84}$ are discussed. The results show that the $B_{\mathrm{ij}}$ matrix element dominates in both transitions. The agreement with the predictions of conserved-vector-current theory is good. Further experiments are suggested which would set better limits on the nuclear-matrix elements.

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