The equations of the title relate thermodynamic functions to correlation-function integrals. They are central to much of statistical mechanics. Their meaning and origin are reviewed here. It is known that the integrals are invariant to the choice of the molecules' centers from which one measures the distances between them. This is illustrated with many examples. A case is discussed in which one of the relations appears to be violated. That is impossible and the origin of the discrepancy is still being sought.