Until recently, general solutions of several classes of equations, such as that which occurs in the theory of the figure of the earth, could not be arrived at. An ingenious transformation lately rendered it a matter of comparative ease to arrive at a solution of these equations in those forms in which they are presented in the solution of physical problems; but still much remains to be done. The object of the present paper is to supply some portion of the deficiency in this respect, by the introduction of a new transformation, and the adoption of the function Γ. The solutions thus obtained are perfectly general, and are arrived at with the greatest facility.