  The argument that Maxwell had given in 1860 to single out this distribution relied on the fact that this is the only probability distribution that is both spherically symmetric and factorizes into functions of the orthogonal components \(v_x, v_y, v_z\) separately. In 1867, however he replaced these desiderata with the more natural requirement that the equilibrium distribution should be stationary, i.e. it should not change shape as a result of the continual collisions between the particles. This called for a more elaborate argument, involving a detailed consideration of the collisions between particles. The crucial assumption in this argument is what is now known as the SZA. Roughly speaking, it states that the number of particle pairs, \(dN(\vec{v}_1, \vec{v}_2)\) with initial velocities between \(\vec{v}_1\) and \(\vec{v}_1 + d^3\vec{v}_1\) and between \(\vec{v}_2\) and \(\vec{v}_2 + d^3\vec{v}_2\) respectively, which are about to collide in a time span \(dt\) is proportional to