  In this article is given a simple expression for the \textit{ center of mass}
for a system of material points in a two-dimensional surface of constant
negative Gaussian curvature. Using basic techniques of Geometry, an expression
in intrinsic coordinates is obtained, and it is showed how it extends the
definition for the Euclidean case. The argument is constructive and also serves
for defining center of mass of a system of particles on the one-dimensional
hyperbolic space $\mathbb{L}^1_R$. Finally, is showed some applications to the
curved $2$- and $3$-body problems.
