The B . W . K . Approximation and Hill ' S Equation , Ii

1. The B.W.K. Method. The B.W.K. procedure1 was discovered in connection with problems of wave mechanics, where Planck's constant h was treated as a very small quantity, and functions could be expanded in series of powers of h. The first terms of the expansion correspond to classical mechanics (completed with some quantum conditions) and higher terms represent typical wave-mechanical effects. Optical problems can be discussed along similar lines, starting with "geometrical optics," and obtaining "physical optics" (diffraction, for instance) as higher order corrections. The B.W.K. method was discussed recently2 by the present author in connection with its possible use in a purely mathematical problem, the discussion of Hill's equation. The method sketched on this occasion proved very valuable and showed the need for a more complete discussion of the whole question. It will be shown that the B.W.K. procedure can yield a very good approximation in a great many mathematical problems, and leads directly to asymptotic expressions similar to those obtained by P. Debye in the case of Bessel functions. There is a variety of problems leading to equations of the general type