Note on the aerodynamic heating of an oscillating insulated surface
of Sec. 6), then Riemann's integral re-presentation of the solution (7) remains valid under the assumptions, (i) and (ii), of Picard's theorem alone. In fact, Riemann's integral representation6 of the solution (7) contains only the following elements: (a) the boundary data <p(x), <f/(y) [which are subject to condition (3)] and their first derivatives and (/3) the function H{x, y; £, r?) and their first derivatives on the boundary; cf. (5), (5 bis). But item (0) does not involve the second derivatives of the function H (derivatives which do in general exist). On the other hand, not only the functions <p(x), tp(y) but also their first derivatives, introduced by item (a), are controlled by assumption (ii). Hence it is clear that the italicized assertion, concerning the general validity of Riemann's formula, can be concluded by the same argument (this time by approximating a(x, y), b(x, y) and <p{x), \p{y) as well as d<p{x)/dx, d\p(y)/dy by sequences of smooth functions) which was applied in Sec. 7.