Page:Elementary Principles in Statistical Mechanics (1902).djvu/129

Rh we get by (336)  Substituting this value in  which expresses the probability that the energy of an unspecified system of the ensemble lies between the limits $$\epsilon'$$ and $$\epsilon''$$, we get  When the number of degrees of freedom is very great, and $$\epsilon-\epsilon_0$$ in consequence very small, we may neglect the higher powers and write

This shows that for a very great number of degrees of freedom the probability of deviations of energy from the most probable value ($$\epsilon_0$$) approaches the form expressed by the 'law of errors.' With this approximate law, we get