Title Criticism of Onsager's reciprocal relations Author Warren D. Smith, NECI Abstract Onsager's second kind of ``reciprocal relations'' %[Phys.Rev. 37 (1931) 405-426; 38 (1931) 2265-2279] state that $\sigma (\vec{H}) = \sigma^{T} (-\vec{H})$ where $\sigma$ is the $3 \times 3$ conductance matrix, $\vec{H}$ is an externally applied magnetic field, and $T$ means matrix transpose. This is only supposed to hold for ``linear'' materials for which the current density $\vec{J}$ is related to the electric field $\vec{E}$ by $\langle \vec{J} \rangle = \sigma \vec{E}$. This is just one instance of a class of mathematically equivalent but physically different relations; e.g. the same relation is supposed to hold if $\sigma$ is thermal conductivity and $\vec{H}$ is angular velocity for a rotating system. We demonstrate by two counterexamples that this relation (and its isomorphs) can be dramatically false. We also have counterexamples to the Onsager relations of the first kind (which are the special case $\vec{H}=\vec{0}$). There are two flaws in Onsager's reasoning and each counterexample exploits one of them. Both flaws had already been thought of %[M.Lax Symmetry Principles in Solid State and Molecular Physics, %Wiley-Interscience 1973] %287-292 but I feel that they are insufficiently appreciated. If we instead define $\sigma$ to be the $3\times 3$ {\em differential conductance} (Jacobian) matrix $\sigma = {\rm d}\langle \vec{J} \rangle/{\rm d} \vec{E}$, then $\sigma_{D} ( \vec{H}, \vec{E} ) = \sigma_{L} ( \vec{H}, \vec{E} )$ where $\sigma_D$ is for the original material, but $\sigma_L$ is for a ``reversed'' material in which all spins and angular momenta (and consequently all magnetic dipoles), and the chirality (by a central inversion of 3-space) are reversed. %This allows nonlinear electrical materials, but %only applies for the $3$ diagonal entries of $\sigma$. In the limit $| \vec{E} | \to 0$ (the ``zero voltage limit'') $\sigma_{D} ( \vec{H} ) = \sigma_{U}^{T} ( -\vec{H} )$ where $\sigma_U$ is for a ``spin-reversed'' material. Note that the $U$ (or $L$) and $D$ materials could be inequivalent -- in which case Onsager's relation is not about one, but actually about two, materials, and hence is less useful. Apparently there have been zero intentional experimental tests (so far) of any Onsager relation in a material inequivalent to its reversal; but I will argue that a 1996 experiment by Solin, Thio, et al. quite likely disproved Onsager's relations of the second kind. %although those authors had not realized it. Keywords Onsager relation, galvanomagnetic effects, magnetoresistance, non-ohmic resistors, nonlinear electrical materials, chirality, errors in textbooks.