Page:Elementary Principles in Statistical Mechanics (1902).djvu/120

96 etc., when $$V_q$$ is a continuous function of $$\epsilon_q$$ commencing with the value $$V_q = 0$$, or when we choose to attribute to $$V_q$$ a fictitious continuity commencing with the value zero, as described on page 90.

If we substitute in these equations the values of $$V_p$$ and $$e^{\phi_p}$$ which we have found, we get  where $$e^{\phi_q} d\epsilon_q $$ may be substituted for $$dV_q$$ in the cases above described. If, therefore, $$n$$ is known, and $$V_q$$ as function of $$\epsilon_q$$, $$V$$ and $$e^\phi$$ may be found by quadratures.

It appears from these equations that $$V$$ is always a continuous increasing function of $$\epsilon$$, commencing with the value $$V=0$$, even when this is not the case with respect to $$V_q$$ and $$\epsilon_q$$. The same is true of $$e^\phi$$, when $$n>2$$, or when $$n=2$$ if $$V_q$$ increases continuously with $$\epsilon_q$$ from the value $$V_q = 0$$.

The last equation may be derived from the preceding by differentiation with respect to $$\epsilon$$. Successive differentiations give, if $$h < \tfrac 12 n + 1$$, $$d^h V/d\epsilon^h$$ is therefore positive if $$h < \tfrac12 n + 1$$. It is an increasing function of $$\epsilon$$, if $$h < \tfrac 12 n$$. If $$\epsilon$$ is not capable of being diminished without limit, $$d^h V/d\epsilon^h$$ vanishes for the least possible value of $$\epsilon$$, if $$h < \tfrac 12 n$$. If $$n$$ is even,