The fundamental notion of statistical mean values in fluid mechanics

stability relative to this special class. Moreover the problem can be extended, within the framework of our general method, by considering obstacles whose contours are composed of a finite number of analytic arcs with shock waves originating at their points of intersection. It is evident that instability at V, or local instability, is sufficient to insure instability in the large. Hence, the above result on instability gives the complete answer to the problem of determining the conditions for instability of shock lines attached to the vertex V of an obstacle whose contour is an analytic curve. Since at most two shock angles a at V are mathematically possible the shock line which actually occurs and which corresponds to the shock line experimentally observed must therefore be the one whose inclination a lies in the interval ao(M) < a < A(M). This may be accepted as sufficient evidence for the stability (local or in the large) of shock lines with inclination a in the interval ao(M) < a < #(M) by those not interested in an existence-theoretic treatment of the problem.