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

Rh linear functions of the $$p$$'s. 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 where $$u_1\ldots u_n$$ are such linear functions of the $$p$$'s. If we write  for the Jacobian or determinant of the differential coefficients of the form $$dp/du$$, we may substitute  for  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