An attempt is made, in the context of the integral representations of the four-point dual amplitudes, to lower the critical degree of space-time dimensions on which dual amplitudes are established. The critical dimension and the intercept of the leading Regge trajectory are determined by an invariancy (under a dual operation) of a differential form (including a volume element), which thus plays a role of measure. A derivation is given of a measure à la Nahm which yields the worldly dimensions in such a way that it appears as a natural generalization of that of the Neveu-Schwarz model, hence still suffering from a presence of tachyon. Its relevancy is studied within the simplest case of two off-shell dual amplitudes. Also specified is the modular group associated with the new volume element.The volume element points to a new pair of the fermi-type dynamical fields, whose construction has not, however, been tried.