When HUYGEN’s principle is applied to the problem of the straight edge, FRESNEL’s diffraction phenomena in the neighbourhood of the geometrical shadow can be accounted for, and the theory agrees closely with observation. But so many approximations are involved in the application of FRESNEL’s theory, that an outstanding event in the history of diffraction theory was the discovery of the exact solution for waves impinging upon a semi-infinite plane. This problem constitutes the only one in diffraction theory which has been solved completely in a comparatively simple form. It is a special case of the wedge problem, the successful treatment of which is due to the fact that there are no dimensions concerned which bear a relation to the wave-length of the incident disturbance. The solution of the problem is due to the labours of a number of mathematicians, among whom POINCARE (‘ Acta Math.,’ vol. 16, p. 297 (1892-3 )), SOMMERFELD (“ Math. Theorie der Diffraction,” ‘ Math. Ann.,’ vol. 47, pp. 317-374 (1895 ) ), MACDONALD (“ Electric Waves,” and ‘ Proc. London Math. Soc.,’ ser. 2 , vol. 14, part 6), and BROMWICH {ibid.) , may be mentioned.