${\mathrm{K}}_{{\mathrm{a}}_1},{\mathrm{K}}_{{\mathrm{a}}_2}\text{ and }{\mathrm{K}}_{{\mathrm{a}}_3}$, are the respective ionisation constants for the following reactions.
$$\begin{gathered} {\mathrm{H}}_2\mathrm{S}\rightleftharpoons {\mathrm{H}}^{+}+{\mathrm{HS}}^{-} \\ {\mathrm{HS}}^{-}\rightleftharpoons {\mathrm{H}}^{+}+{\mathrm{S}}^{2-} \\ {\mathrm{H}}_2\mathrm{S}\rightleftharpoons 2{\mathrm{H}}^{+}+{\mathrm{S}}^2 \end{gathered}$$
The correct relationship between ${\mathrm{K}}_{{\mathrm{a}}_1},{\mathrm{K}}_{{\mathrm{a}}_2}\text{ and }{\mathrm{K}}_{{\mathrm{a}}_3}$ is

$$\begin{gathered} \text{ For the reaction, }{\mathrm{H}}_2\mathrm{S}\rightleftharpoons {\mathrm{H}}^{+}+{\mathrm{HS}}^{-} \\ {\mathrm{K}}_{{\mathrm{a}}_1}\rightleftharpoons \frac{\left[{\mathrm{H}}^{+}\right]\left[{\mathrm{HS}}^{-}\right]}{\left[{\mathrm{H}}_2\mathrm{S}\right]} \\ \text{ For the reaction, }{\mathrm{HS}}^{-}\rightleftharpoons {\mathrm{H}}^{+}+{\mathrm{S}}^{2-};{\mathrm{K}}_{{\mathrm{a}}_2}=\frac{\left[{\mathrm{H}}^{+}\right]\left[{\mathrm{S}}^{2-}\right]}{\left[{\mathrm{HS}}^{-}\right]} \end{gathered}$$
When the above two reactions are added, their equilibrium constants are multiplied, thus

$$\begin{gathered} {\mathrm{K}}_{{\mathrm{a}}_3}=\frac{{\left[{\mathrm{H}}^{+}\right]}^2\left[{\mathrm{S}}^{2-}\right]}{\left[{\mathrm{H}}_2\mathrm{S}\right]}={\mathrm{K}}_{{\mathrm{a}}_1}\times {\mathrm{K}}_{{\mathrm{a}}_2} \\ \mathrm{Hence} {\mathrm{K}}_{{\mathrm{a}}_3}={\mathrm{K}}_{{\mathrm{a}}_1}\times {\mathrm{K}}_{{\mathrm{a}}_2} \end{gathered}$$