The atomic beams magnetic resonance method has been applied to the study of the h.f.s. of the metastable $^{2}P_{\frac{3}{2}}$ state of both ${\mathrm{In}}^{113}$ and ${\mathrm{In}}^{115}$. It has been found that the separations of the energy levels at zero external magnetic field are completely described by the equation $W=\frac{\mathrm{haC}}{2}+hbC(C+1)$, where $C=F(F+1)\ensuremath{-}I(I+1)\ensuremath{-}J(J+1)$, and $a$ and $b$ are the magnetic dipole and electric quadrupole interaction constants, respectively. No evidence has been found to indicate the existence of a nuclear moment of higher order than the quadrupole moment. These are the heaviest atoms in which a critical search for such moments has been made.The numerical values of the constants for ${\mathrm{In}}^{115}$ are: ${{a}^{115},=242.165\ifmmode\pm\else\textpm\fi{}0.002 \frac{\mathrm{mc}}{sec}.}{{b}^{115},=1.56098\ifmmode\pm\else\textpm\fi{}0.00006 \frac{\mathrm{mc}}{sec}.}$ The ratio $\frac{{a}^{115}}{{a}^{113}}$ has been determined by Hardy and Millman to be 1.00224\ifmmode\pm\else\textpm\fi{}0.00010. From this value, and the results of our measurements, the numerical values of the constants for ${\mathrm{In}}^{113}$ are: ${{a}^{113},=241.624\ifmmode\pm\else\textpm\fi{}0.024 \frac{\mathrm{mc}}{sec}.}{{b}^{113},=1.53855\ifmmode\pm\else\textpm\fi{}0.00015 \frac{\mathrm{mc}}{sec}.}$ where the uncertainties in ${a}^{113}$ and ${b}^{113}$ follow from the uncertainty in the ratio of the $a'\mathrm{s}$. The numerical values of the electric quadrupole moments are: ${{Q}^{115},=1.161\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}24}{\mathrm{cm}}^{2}}{{Q}^{113},=1.144\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}24}{\mathrm{cm}}^{2}.}$ It is seen from the ratios $\frac{{I}^{115}}{{I}^{113}}=1$, $\frac{{a}^{115}}{{a}^{113}}=1.00224$, $\frac{{b}^{115}}{{b}^{113}}=1.01146$, that in these measurable nuclear properties ${\mathrm{In}}^{113}$ and ${\mathrm{In}}^{115}$ exhibit the greatest degree of similarity observed for any pair of isotopes of odd mass differing in mass by two units.