On the Existence of a Wave of Greatest Height and Stokes's Conjecture

It is shown that there exists a solution of Nekrasov's integral equation which corresponds to the existence of a wave of greatest height and of permanent form moving on the surface of an irrotational, infinitely deep flow. It is also shown that this wave is the uniform limit, in a specified sense, of waves of almost extreme form. The question of the validity of Stokes's conjecture is reduced to one of the regularity of the solution of Nekrasov's equation in this limiting case.