Page:Elementary Principles in Statistical Mechanics (1902).djvu/75

linear functions of the $p$'s.^[1] The coefficients in these linear functions, like those in the quadratic function, must be regarded in the general case as functions of the $q$'s. Let

$2\epsilon _{p}=u_{1}{}^{2}+u_{2}{}^{2}\ldots +u_{n}{}^{2}$

where $u_{1}\ldots u_{n}$ are such linear functions of the $p$'s. If we write

${\frac {d(p_{1}\ldots p_{n})}{d(u_{1}\ldots u_{n})}}$

for the Jacobian or determinant of the differential coefficients of the form $dp/du$, we may substitute

${\frac {d(p_{1}\ldots p_{n})}{d(u_{1}\ldots u_{n})}}\,du_{1}\ldots du_{n}$

for

$dp_{1}\ldots dp_{n}$

under the multiple integral sign in any of our formulæ. It will be observed that this determinant is function of the $q$'s alone. The sign of such a determinant depends on the relative order of the variables in the numerator and denominator. But since the suffixes of the $u$'s are only used to distinguish these functions from one another, and no especial relation is supposed between a $p$ and a $u$ which have the same suffix, we may evidently, without loss of generality, suppose the suffixes so applied that the determinant is positive. Since the $u$'s are linear functions of the $p$'s, when the integrations are to cover all values of the $p$'s (for constant $q$'s) once and only once, they must cover all values of the $u$'s once and only once, and the limits will be $\pm \infty$ for all the $u$'s. Without the supposition of the last paragraph the upper limits would not always be $+\infty$, as is evident on considering the effect of changing the sign of a $u$. But with the supposition which we have made (that the determinant is always positive) we may make the upper limits $+\infty$ and the lower $-\infty$ for all the $u$'s. Analogous considerations will apply where the integrations do not cover all values of the $p$'s and therefore of

1. The reduction requires only the repeated application of the process of 'completing the square' used in the solution of quadratic equations.