According to \textit{I. M. James} [The topology of Stiefel manifolds, Lond. Math. Soc. Lect. Note Ser. 24 (1976; Zbl 0337.55017)], the quaternionic quasi-projective space \({\mathbb{H}}{\mathbb{Q}}_ n\) is defined in two ways. In this paper the authors show that the two definitions are equivalent and that the map \(t_ n: {\mathbb{H}}{\mathbb{Q}}_ n\to E({\mathbb{C}}{\mathbb{P}}_+^{2n- 1})\) of \textit{H. Toda} and \textit{K. Kozima} [J. Math. Kyoto Univ. 22, 131- 153 (1982; Zbl 0502.55004)] can be identified with the connecting map in the cofibre sequence starting with \(g_{n+}: {\mathbb{C}}{\mathbb{P}}_+^{2n- 1}\to {\mathbb{H}}{\mathbb{P}}_+^{n-1}\). Here, \(X_+=X\cup \{one\) point\(\}\) and \(g_{n+}\) stands for the map induced from the standard fibration \(g_ n: {\mathbb{C}}{\mathbb{P}}^{2n-1}\to {\mathbb{H}}{\mathbb{P}}^{n-1}\). The authors also construct a natural mapping from \(E({\mathbb{H}}{\mathbb{P}}_+^{n-1})\) to \(X_ n=U(2n)/Sp(n)\) by use of the first result. As an application, the authors determine the order of the attaching class \(E\gamma_{n- 1}({\mathbb{H}})\in \pi_{4n}(E{\mathbb{H}}{\mathbb{P}}^{n-1})\) for even n. [Dr. Shichirô Oka died on 30th October, 1984.]