The various formulae for the Legendre Functions, and the relations between these formulae, have been studied by Kummer,Riemann, Olbricht, Hobson, Barnes, Whipple, and others. Hobson obtained some of the relations directly, by expres ing the functions as Pochhammer integrals, and expanding in a number of series each with its own region of convergence. To obtain some of the other formulae, such as (i) below, he transformed the differential equation, and then expressed the functions in terms of the solutions of the transformed equation. Barnes succeeded, by means of his wellknown integrals involving Gamma Functions, in deducing all the formulae directly from the formulae which define the functions.Notes on the history of the subject and references to previous work will be found in the papers by Hobson and Barnes.