Example 4 Given $a+x^{2}=1991, b+x^{2}=$ $1992, c+x^{2}=1993$, and $a b c=24$. Find the value of the algebraic expression $\frac{a}{b c}+\frac{b}{a c}+\frac{c}{a b}-\frac{1}{a}-\frac{1}{b}-\frac{1}{c}$.
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Solution: Eliminating $x^{2}$ from the known conditions, we get $a-b=$
$$
\begin{array}{l}
-1, b-c=-1, a-c=-2 . \text { Then } \\
\text { the original expression }=\frac{a^{2}+b^{2}+c^{2}-a b-b c-a c}{a b c} \\
= \frac{(a-b)^{2}+(b-c)^{2}+(a-c)^{2}}{2 a b c} \\
= \frac{(-1)^{2}+(-1)^{2}+(-2)^{2}}{2 \times 24}=\frac{1}{8} .
\end{array}
$$
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