10. Let $a_{1}, a_{2}, \cdots, a_{n}$ be given real numbers, not all zero, and $r_{1}, r_{2}, \cdots, r_{n}$ be real numbers. If the inequality $r_{1}\left(x_{1}-a_{1}\right)+r_{2}\left(x_{2}-a_{2}\right)+\cdots+r_{n}\left(x_{n}-a_{n}\right) \leqslant$ $\sqrt{x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}}-\sqrt{a_{1}^{2}+a_{2}^{2}+\cdots+a_{n}^{2}}$ holds for any real numbers $x_{1}, x_{2}, \cdots, x_{n}$, find the values of $r_{1}, r_{2}, \cdots, r_{n}$.
10. From the problem, for any real numbers $x_{1}, x_{2}, \cdots, x_{n}$, the inequality 
$$
r_{1}\left(r_{1}-a_{1}\right)+r_{2}\left(x_{2}-a_{2}\right)+\cdots+r_{n}\left(x_{n}-a_{n}\right) \leqslant \sqrt{x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}}-\sqrt{a_{1}^{2}+a_{2}^{2}+\cdots+a_{n}^{2}}
$$
holds. 
By setting $x_{k}=2 a_{k}$ and $x_{k}=0 (k=1,2, \cdots, n)$ in the above inequality, we have
$$
\begin{array}{l}
r_{1} a_{1}+r_{2} a_{2}+\cdots+r_{n} a_{n} \leqslant \sqrt{a_{1}^{2}+a_{2}^{2}+\cdots+a_{n}^{2}} . \\
r_{1} a_{1}+r_{2} a_{2}+\cdots+r_{n} a_{n} \geqslant \sqrt{a_{1}^{2}+a_{2}^{2}+\cdots+a_{n}^{2}} .
\end{array}
$$

Thus, 
$$
r_{1} a_{1}+\dot{r}_{2} a_{2}+\cdots+r_{n} a_{n}=\sqrt{a_{1}^{2}+a_{2}^{2}+\cdots+a_{n}^{2}}.
$$
Therefore, 
$$
r_{1} x_{1}+r_{2} x_{2}+\cdots+r_{n} x_{n} \leqslant \sqrt{x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}}.
$$

By the Cauchy-Schwarz inequality, we have
$$
r_{1} a_{1}+r_{2} a_{2}+\cdots+r_{n} a_{n} \leqslant \sqrt{a_{1}^{2}+a_{2}^{2}+\cdots+a_{n}^{2}} \cdot \sqrt{r_{1}^{2}+r_{2}^{2}+\cdots+r_{n}^{2}}.
$$

This implies that 
$$
1 \leqslant \sqrt{r_{1}^{2}+r_{2}^{2}+\cdots+r_{n}^{2}}.
$$
Since the inequality $(**)$ holds for any real numbers $x_{1}, x_{2}, \cdots, x_{n}$, setting $x_{k}=r_{k} (k=1,2, \cdots, n)$, we get 
$$
\sqrt{r_{1}^{2}+r_{2}^{2}+\cdots+r_{n}^{2}} \leqslant 1.
$$
Thus, 
$$
r_{1}^{2}+r_{2}^{2}+\cdots+r_{n}^{2}=1,
$$
and hence,
$$
r_{1} a_{1}+r_{2} a_{2}+\cdots+r_{n} a_{n}=\sqrt{a_{1}^{2}+a_{2}^{2}+\cdots+a_{n}^{2}} \cdot \sqrt{r_{1}^{2}+r_{2}^{2}+\cdots+r_{n}^{2}}.
$$

Therefore, there exists a real number $t$ such that 
$$
r_{k}=t a_{k} (k=1,2, \cdots, n),
$$
which implies
$$
t\left(a_{1}^{2}+a_{2}^{2}+\cdots+a_{n}^{2}\right)=\sqrt{a_{1}^{2}+a_{2}^{2}+\cdots+a_{n}^{2}},
$$
and solving for $t$ gives 
$$
t=\frac{1}{\sqrt{a_{1}^{2}+a_{2}^{2}+\cdots+a_{n}^{2}}}.
$$
Thus, 
$$
r_{k}=\frac{a_{k}}{\sqrt{a_{1}^{2}+a_{2}^{2}+\cdots+a_{k}^{2}}} (k=1,2, \cdots, n).
$$