  The concept of centre of mass of two particles in 2D spaces of constant
Gaussian curvature is discussed by recalling the notion of "relativistic rule
of lever" introduced by Galperin [Comm. Math. Phys. 154 (1993), 63--84] and
comparing it with two other definitions of centre of mass that arise naturally
on the treatment of the 2-body problem in spaces of constant curvature: firstly
as the collision point of particles that are initially at rest, and secondly as
the centre of rotation of steady rotation solutions. It is shown that if the
particles have distinct masses then these definitions are equivalent only if
the curvature vanishes and instead lead to three different notions of centre of
mass in the general case.
