On Professor Whittaker's solution of differential equations by definite integrals: Part II

Abstract

In Part I it has been shown that, given a contact transformation, two equationscan be derived which lead to the compatible differential equationsIt will be shown in the present communication that the necessary and sufficient condition that (1.3), (1.4) should be compatible is thatregarded as an equation in the non-commutative variables q, p which themselves satisfy the conditionWe shall call functions satisfying this condition conjugate functions. From this point of view the method employed by Professor Whittaker in his original paper, involving the use of a contact transformation,, was really a particular method of generating conjugate functions. This powerful method may be supplemented and extended by the other methods developed in the following pages.