Abstract: Despite the great significance of equations with doubly-periodic coefficients in the methods of mathematical physics, the problem of solving Lamb-Helmholtz equation still remains to be tackled; Arscott and Möeglich's method of double-series expansion as well as Malurkar's non-linear integral equation are unable to reach the final solution, Our main result consists in obtaining analytic expression for ellipsoidal wave funcdons of four species ε_ci(sna),ε_si(sna)(i=1,2,3,4) by deriving a couple of linear integral equations and solving these by integral transform, including the well-known Lamé function E_ci(sna),E_si(sna) as special case.Generalizing Riemann's idea of P-function, we introduce D-function to eapress their transformation properties.