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In this note, F denotes the field of the complex numbers C or the field of the quaternions H. We denote by FP the F-projective space of nF-dimensions and by Qn(F) the quasi-F-projective space. Gn(F) denotes the unitary group U(ri) or the symplectic group Sp(ri) according as F is C or H. Let d be the dimension of F over the field of the real numbers R and S~ the unit sphere in F". Let T'n: S ~—> Gn-i(F) be the characteristic map for the normal form of the principal Gn_i(F)-bundle over S ~. Then, as is well known ([2], [3] and [9]), Ini T^Q^-^F), precisely, the following diagram commutes:

The order of the attaching class in the suspended quaternionic quasiprojective space