A sample of $K^{+}$ mesons stopped in the CERN 1.1-${\mathrm{m}}^3$ heavy-liquid bubble chamber has been used to study the Dalitz-plot density and the ${\mu }^{+}$ polarization in ${K_{\mu 3}}^{+}$ decay and to measure the ${K_{\mu 3}}^{+}$ and ${K_{e3\mathrm{}}}^{+}$ branching ratios. All three analyses are described in detail, and they are used, separately and combined, to study the form of the strangeness-changing, semileptonic weak interaction. This is found to be consistent with a coupling of vector currents and to be invariant under time reversal. The vector form factors are studied with a wide range of assumptions as to their possible four-momentum ($q^2$) dependence, and the results of all three measurements are in agreement. If ${\lambda }_{+}$ is fixed at 0.029 (the average value from ${K_{e3\mathrm{}}}^{+}$ experiments) and if $\xi$ is assumed to be independent of $q^2$, then the result is $\xi =-0.65\mathrm{}\pm 0.13$. Assuming that $f+\left(q^2\right)$ and $\xi \left(q^2\right)$ are both linear functions of $q^2$, a three-parameter fit gives ${\lambda }_{+}=0.060\mathrm{}\pm 0.019$, $\xi \mathrm{}\left(0\right)=-1.0\mathrm{}\pm 0.5$, and $\xi \left(5.1\mathrm{}{m_{\pi }}^2\right)=-0.97\mathrm{}\pm 0.20$.

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