The author proves some necessary existence conditions for quasi-symmetric designs, i.e. designs with exactly two block intersection sizes. Properties of the incidence matrix of the design are used to achieve the results which rule out several sets of admissible parameters for quasi- symmetric designs. Consequences of the shown results on the structure of the strongly regular graph associated with the design are proved too. This paper should be read together with the paper by \textit{A. Blokhuis} and \textit{A. R. Calderbank} [ibid. 189-206 (1992; see the review below)] as both papers deal with the same topic and share some results proved by similar techniques.