Why doesn't N2O give a positive brown ring test?
The brown ring nitrate test is based on forming ferrous nitrosyl cation, with nitrous oxide formed from nitrates by iron oxidation in acidic solution:

$$\begin{align}
\ce{3 [Fe(H2O)6]^2+ + NO3- + 4 H+ &amp;-&gt;3 [Fe(H2O)6]^3+ + NO + 2 H2O} \\
\ce{[Fe(H2O)6]^2+ + NO &amp;-&gt;3 [Fe(H2O)(NO)]^2+} + H2O\\
\end{align}$$

Positive results are given also by nitrites and nitrogen oxides able to form nitric or nitrous acid, directly by hydration, or indirectly by disproportionation.

$$\begin{align}
\ce{ [Fe(H2O)6]^2+ + NO2- + 2 H+ &amp;-&gt; [Fe(H2O)6]^3+ + NO +  H2O} \\
\ce{N2O3 + H2O &amp;-&gt; 2 HNO2}\\
\ce{2 NO2 + H2O &amp;-&gt; HNO2 + HNO3}\\
\ce{N2O5 + H2O &amp;-&gt; 2 HNO3}\\
\end{align}$$

Therefore, $\ce{NO}$ and all higher nitrogen oxides oxidizing $\ce{Fe^2+}$, forming $\ce{NO}$, react positively in the brown ring test. 

$\ce{N2O}$ cannot do that, because it cannot form $\ce{NO}$ at given conditions.