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N. G. van Kampen [5] evaluated asymptotically the integrals given by the modified Kirchhoff method. His result also contains one stationary point corresponding to each edge-diffracted ray through P, and in addition one stationary point corresponding to each corner of the edge, accounting for the corner-diffracted rays. R. M. Lewis, B. D. Seckler, and the present author [6] have obtained similar results from W. Braunbek's [7] modification of the Kirchhoff theory. 50 JOSEPH B. KELLER Surface rays appear in the asymptotic expansion for large ka of the field diffracted by a sphere or cylinder of radius a.

M. Lewis, and B. Seckler [33], J. B. Keller [24,29,34,35], K. 0. Friedrichs and J. B. Keller [36], B. R. Levy and J. B. Keller [37], S. N. Karp and J. B. Keller [38], B. D. Seckler and J. B. Keller [40], etc. Whenever possible, the fields constructed by the ray method were compared A GEOMETRICAL THEORY OF DIFFRACTION 51 with asymptotic expansions (for large k) of exact solutions. In all such cases perfect agreement was obtained. In other cases numerical results were compared, and good agreement was obtained for ka >_ 2, where a is a typical length in the problem.

M. Kline, An asymptotic solution of Maxwell's equations, Comm. Pure Appl. Math. vol. IV no. 2-3 (August, 1951), pp. 225-263. Asymptotic solution of linear hyperbolic partial differential equations, J. Rational Mech. Anal. vol. 3 no. 3 (May, 1954). 20. F. G. Friedlander, Geometrical optics and Maxwell's equations, Proc. Cambridge Philos. Soc. vol. 43 part 2 (1946) pp. 284-286. 21. H. Bremmer, The jumps of discontinuous solutions of the wave equation, Comm. Pure Appl. Math. vol. IV no. 4 (November, 1951) pp.

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