We reexamine the interpretation of the normal force measured with a cone-and-plate rheometer for nematic liquid crystals. We point out the fact that the quite widespread belief that the normal force is directly related to the well-known rheological function N-1 (i.e., the first normal stress difference) fails for these complex fluids. After a brief presentation of the theoretical bases leading to the general expression of the normal force, this new approach is applied to nematic liquid crystals within the framework of Leslie-Ericksen theory. In order to avoid heavy numerical computations the full Leslie-Ericksen equations describing the three-dimensional flow within the cone-and-plate cell are reduced, thanks to reasonable approximations, to an effective one-dimensional problem (i.e., a system of partial differential equations with spatial derivatives with respect to one single coordinate). Results for low molecular weight liquid crystals are presented and discussed. The main feature evidenced in this work is the notable difference between the normal force calculated in this way and the N-1 function, in particular the normal force function becomes negative for high shear rate while N-1 stays positive.