  By integrating this expression over all momenta but one, and dividing this by the integral of \(\rho_{mc}\) over all momenta, Boltzmann obtained the marginal probability density \(\rho_{mc}(\vec{p}_1 \mid \vec{q}_1,\ldots,\vec{q}_N)\) for particle 1's momentum, conditionalized on the particle positions \(\vec{q}_1,\ldots,\vec{q}_N\).  He then showed that this marginal probability distribution tends to the Maxwell distribution when the number of particles tends to infinity.