The literature concerning these equations being very extensive, we shall not attempt to give a complete list of references. The starting point for many more modern researches has been the work of S. Bernstein,t who was the first to prove the analyticity of the solutions of the general equation with 4 analytic and who was able to obtain a priori bounds for the second and higher derivatives of z in the quasi-linear type in terms of the bounds of |z I | I p I , I q I and the derivatives of the coefficients. He was also able to prove the existence of the solution of the quasi-linear equation in some very general cases. He assumed that all the data were analytic. However, his papers are very complicated and certain details require modification. On account of the results of J. Horn, L. Lichtenstein, and many others,: the restriction of analyticity has been removed. Some very interesting modern work has been done by Leray and Schauder? in a paper in which they develop a general theory of nonlinear functional equations and apply their results to quasi-linear equations.