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The explicit formula for the hyperbolic metric $\lambda_{\alpha,\,\beta,\,\gamma}(z)|dz|$ on the thrice-punctured sphere $\mathbb{P} \backslash \{z_1,\,z_2,\,z_3\}$ with singularities of order $\alpha,\,\beta,\,\gamma \leq 1$ with $\alpha+\beta+\gamma>2$ at $z_1,\,z_2,\,z_3$ was given by Kraus, Roth and Sugawa in \cite{Rothhyper}. In this paper we investigate the asymptotic properties of the higher order derivatives of $\lambda_{\alpha,\,\beta,\,\gamma}(z)$ near the singularity and give some more precise description for the asymptotic behavior.

ASYMPTOTIC PROPERTIES OF THE HYPERBOLIC METRIC ON THE SPHERE WITH THREE CONICAL SINGULARITIES