If i/ be taken as the stream function for the flow of the Tricomi gas, the stream lines i1 = const. in the (0, a)-plane (hodograph plane) constitute a one-parameter family of straight lines enveloping the semicubic parabola 902 = -4u3. This semi-cubic parabola is one of the characteristics of Tricomi's equation. Within the region 902< -_ the solution is triple-valued and the solution surface ,6 = 6(0, a) consists of three sheets, two of which unite along curves which project onto the characteristic 902 = 43 in the (0, a)-plane. Thus this characteristic serves as the locus of branch points for the algebraic function 4I(0, a). If Tricomi's equation be replaced by a linear equation