Abstract
Lame's differential equation arises when the wave equation is separated in ellipsoidal or sphero-conal coordinates. Throughout the work which follows, Lame's equation has been used exclusively in its Jacobian elliptic form. Chapter I of Part 1 extends the knowledge of integral relations involving Lame functions of the first and second kinds. Chapter II considers the characteristic solutions of Lame's equation when nu is half an odd integer (the Lame-Wangerin functions). Bounds are determined for the characteristic values of h and a further note gives bounds for the values of h associated with the Lame polynomials. The solution of Lame's equation when nu = 1/2 and h = 1/4 (1+k[2]) is then obtained by direct integration. In the final sections of the Chapter, new forms of the Lame-Wangerin functions are introduced and there is a discussion of their orthogonality properties. In Chapter III it is shown that Lame's equation can be reduced to the hypergeometric equation in various ways if k[2] = 1/2 and h is a certain function of nu. Part 2 is devoted to the consideration of a new perturbation technique which makes use of the Fourier series for sn2 (u,k). The method is applied to some triangular plate problems and there is some comment on further possible applications. The Author wishes to express his great appreciation of the encouragement and valuable criticism given by Professor F.M. Arscott throughout the preparation of this thesis.