  Reciprocal relations correlate fairly accurately a great variety of
experimental results. Nevertheless, the concepts of statistical fluctuations,
and microscopic reversibility - the bases of the accepted proof of the
relations by Onsager - are illusory and faulty, and contradict the foundations
of the science of thermodynamics. The definitions, postulates, and main
theorems of thermodynamics are briefly presented. It is shown beyond a shadow
of a doubt that thermodynamics is a nonstatistical science that applies to all
systems (both macroscopic, and microscopic, including systems that consist
either of only one structureless particle, or only one spin), to all states
(both thermodynamic or stable equilibrium, and not stable equilibrium), and
that includes entropy as a well defined, intrinsic, nonstatistical property of
any system in any state, at any instant in time. In the light of this novel
conception of thermodynamics, we find that reciprocal relations result from a
well known mathematical theorem, to wit, given a well behaved analytic function
of many variables then the second derivative of the function with respect to
any two variables is independent of the order of differentiation, namely,
whether the first derivative is taken with respect to the one or the other of
the two variables.
