  Nonlinear optical properties in the fullerene C$_{60}$ and the extracted
higher fullerenes -- C$_{70}$, C$_{76}$, C$_{78}$, and C$_{84}$ -- are
theoretically investigated by using the exciton formalism and the
sum-over-states method. We find that off-resonant third order susceptibilities
of higher fullerenes are a few times larger than those of C$_{60}$. The
magnitude of nonlinearity increases as the optical gap decreases in higher
fullerenes. The nonlinearity is nearly proportional to the fourth power of the
carbon number when the onsite Coulomb repulsion is $2t$ or $4t$, $t$ being the
nearest neighbor hopping integral. This result, indicating important roles of
Coulomb interactions, agrees with quantum chemical calculations of higher
fullerenes.
