The classical Jacobi formula for the elliptic integrals (Gesammelte Werke I, p. 235) shows a relation between Jacobi theta constants and periods of ellptic curves $E(\lambda): w^2=z(z-1)(z-\lambda)$ . In other words, this formula says that the modular form $\vartheta^{4} _{00}(\tau)$ with respect to the principal congruence subgroup $\Gamma (2)$ of $PSL(2,\mbi{Z})$ has an expression by the Gauss hypergeometric function $F(1/2, 1/2, 1; 1-\lambda)$ via the inverse of the period map for the family of elliptic curves $E(\lambda)$ (see Theorem 1.1). In this article we show a variant of this formula for the family of Picard curves $C(\lambda_1,\lambda_2): w^3=z(z-1)(z-\lambda_1)(z-\lambda_2)$ , those are of genus three with two complex parameters. Our result is a two dimensional analogy of this context. The inverse of the period map for $C(\lambda_1,\lambda_2)$ is established in [S] and our modular form $\vartheta_0^3(u,v)$ (for the definition, see (2.7)) is defined on a two dimensional complex ball $\mathcal{D}=\{ 2{\mathrm{Re}}v+|u|^2<0\}$ , that can be realized as a Shimura variety in the Siegel upper half space of degree 3 by a modular embedding. Our main theorem says that our theta constant is expressed in terms of the Appell hypergeometric function $F_1(1/3, 1/3, 1/3, 1; 1-\lambda_1,1-\lambda_2)$ .