§ 1. Introduction. Let R and S be two ultrahyperelliptic surfaces defined by two equations y* = G(z) and u 2 =g(w), respectively, where G and g are two entire functions having no zero other than an infinite number of simple zeros. Then one of the authors [6], [7] established the following perfect condition for the existence of analytic mappings from R into S. THEOREM A. // there exists an analytic mapping φ from R into S, then there exists a pair of two entire functions h(z) and f(z) satisfying

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... an equation and vice versa. Let yR(R) be a family of non-constant meromorphic functions on R. Let / be a member of 9R(R). Let P(f) be the number of Picard's exceptional values of /, which we say α a Picard's value of / when a is not taken by / on R. Let P(R) be a quantity defined by (cf. [4]). Let P(S) be the corresponding quantity attached to S. In the present paper we shall give a perfect condition for the existence of analytic mappings from R into S in a case of P(7?)=P(S)=4, which is more direct than theorem A, and shall give some characterizations of the surfaces R with P(jR)=3 by the forms of defining functions G. By a characterization, which was given in [5], of R with P(R)=4 by G(z) we can put with two suitable entire functions F and // and two constants γ and δ. Similarly if jP(S)=4, we can put f(wfg(w) = (e L ^ -γ')(e L ^-d'\ L(w) ^ const., (L2) L(0)=0, γ'δ'(γ'-