4・132 There are three unknowns $x, y, z$ satisfying the following equations
$$\left\{\begin{array}{l}
x+y+z=a \\
x^{2}+y^{2}+z^{2}=b^{2} \\
x^{-1}+y^{-1}+z^{-1}=c^{-1}
\end{array}\right.$$

Determine the value of $x^{3}+y^{3}+z^{3}$.
[Solution] (1) ${ }^{2}-(2)$ gives
$$x y+y z+z x=\frac{1}{2}\left(a^{2}-b^{2}\right) .$$
(3) gives $x y z=c(x y+y z+z x)=\frac{c\left(a^{2}-b^{2}\right)}{2}$.

Therefore, $x^{3}+y^{3}+z^{3}$
$$\begin{array}{l}
=(x+y+z)^{3}-3(x+y+z)(x y+y z+z x)+3 x y z \\
=a^{3}-3 a \cdot \frac{1}{2}\left(a^{2}-b^{2}\right)+3 \cdot \frac{c\left(a^{2}-b^{2}\right)}{2} \\
=a^{3}+\frac{3}{2}\left(a^{2}-b^{2}\right)(c-a) .
\end{array}$$