14. Find all positive numbers $a$ such that the quadratic equation $\left(a^{2}+1\right) x^{2}+2 a x+\left(a^{2}-1\right)=0$ has both roots as integers.
$\begin{array}{l}\quad \text { (Hint: }\left(x^{2}+1\right) a^{2}+2 x a+\left(x^{2}-1\right)=0, \Delta=4 x^{2} \\ -4\left(x^{2}+1\right)\left(x^{2}-1\right) \geqslant 0, x^{4}-x^{2}-1 \leqslant 0,0 \leqslant x^{2} \leqslant \\ \frac{1+\sqrt{5}}{2}<2,-\sqrt{2}<x<\sqrt{2}, \therefore x=-1,0,1 \cdot a \\ =1)\end{array}$