  Within the framework of the effective potential theory of the structural
glass transition, we calculate for the p-spin model a static nonlinear
susceptibility related to a four-point density correlation function, and show
that it grows and diverges in mean field with exponent $\gamma=1/2$ as the mode
coupling critical temperature T_c is approached from below. When T_c is
approached from above, we calculate within the mode coupling framework a
dynamic nonlinear susceptibility and show that there is a characteristic time
where the susceptibility is a maximum, and that this time grows with decreasing
T. We find that this susceptibility diverges as T_c is approached from above,
and has key features in common with the ``displacement-displacement
susceptibility'' recently introduced to measure correlated particle motion in
simulations of glass-forming liquids.
