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Consider a parabolic end Ω of a Riemann surface in the sence of Heins [2] (cf. Nakai [3]). A density P = P(z)dxdy (z -x + iy) is a 2-form on Ω = Ω U dΩ with nonnegative locally Holder continuous coefficients P(z). A density P is said to be finite if the integralThe elliptic dimension of a density P at the ideal boundary point δ, dim P in notation, is defined (Nakai [5], [6]) to be the 'dimension' of the half module of nonnegative solutions of the equationon an end Ω with the vanishing boundary values on dΩ. The elliptic dimension of the particular density P = 0 at δ is called the harmonic dimension of δ. After Bouligand we say that the Picavd principle is valid for a density P at δ if dimP = l. For the punctured disk V: 0 < \z\ < 1, Nakai [6] showed that the Picard principle is valid for any finite density P on 0 < \z\ <; 1 at the ideal boundary 2 = 0, and he conjectured that the above theorem is valid for every general end of harmonic dimension one. The purpose of this paper is to give a partial answer in the affirmative.Heins [2] showed that the harmonic dimension of the ideal boundary δ of an end is one if Ω satisfies the condition [H]: There exists a sequence {A n } of disjoint annuli with analytic Jordan boundaries on Ω satisfying the condition that for each n,A n+1 separates A n from the ideal boundary, and A 1 separates the relative boundary dΩ from the ideal boundary? and