Measurements of the first and second normal stress differences of an extremely shear thickening polymer dispersion have been performed at imposed shear stress both by means of a Rheometrics RDS and a home-made Normal Force Cell using cone-plate geometry. The sample is a 58.7 vol.% dispersion of styrene/ethyl acrylate copolymer particles with 280 nm average diameter in glycol. The spheres are electrostatically stabilized by negative charges due to carboxylic groups. In the regime of strong shear thickening a negative first normal stresses difference N-1 is found, the absolute value of which is equal to the absolute value of the shear stress tau : N-1 = -\tau\. Measurements of the normal force acting on a disk of smaller radius than the full plate radius yield for the second normal stress difference N-2 the relation N-2 = -N-1/2. The experimentally observed relation of the apparent shear stress tau(a) and apparent first normal stress difference N-1a in plate-plate geometry, N-1a = -\tau(a)\, can be explained by assuming that only the sample part near the rim is shear thickening whereas the inner part remains in the low viscosity regime. In the latter regime N-1a is positive and smaller than tau(a). It is further explained why small deviations of the concentricity in Searle geometry create a strong lateral force that pulls the bob towards the cup.