We show that there are inflationary models for which perturbations in the energy momentum tensor, which are of second order in the scalar field, cannot be neglected. We first specify the conditions under which the usual first order perturbations are absent. We then analyze classically, the growth and decay of our new type of perturbations for one mode of fluctuations \de\phi k in the scalar field. We generalize this analysis, considering the contribution from the whole spectrum of \de\phi to a given wavelength of geometrical perturbations. Finally, we discuss the evolution of the perturbations during the subsequent radiation dominated era and discuss the resulting spectrum of density fluctuations. In the case of a massless scalar field we find a spectral index n=4. For massive scalar fields we obtain n=0 but the resulting amplitude of fluctuations for inflation around a GUT scale are by far too high. Hence, "conventional" inflationary models must not be influenced by this new type of perturbations, in order to lead to acceptable matter and CMB fluctuations.