Distribution of a studentised difference of two observations

Let $X_1,…,X_n,X_{n+1},X_{n+2},…,X_{n+m}$ ($n>4$, $m>4$) be a random sample from $N(\mu,\sigma^2)$; $\mu\in\mathbb R$, $\sigma>0$. If $\overline X_1=\frac1n\sum_{i=1}^nX_i$ and $\overline X_2=\frac1{m-2}\sum_{i=n+1}^{n+m-2}X_i$, then the distribution of the random variable

$$T=\frac{X_{n+m}-X_{n+m-1}}{\sqrt{\sum_{i=1}^n(X_i-\overline X_1)^2+\sum_{i=n+1}^{n+m-2}(X_i-\overline X_2)^2}}$$

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