The different internal motions of the gas with total energy E define a sphere, the surface of a 3 N -dimensional ball with radius 2 m E . The volume of the sphere is 2 π π 3 N / 2 ( 2 m E ) ( 3 N − − 1 ) / 2 Γ Γ ( 3 N / 2 ) , where Γ Γ is the gamma function.