General comment: despite the longish historical introduction this question is not about the history of physics but rather about a specific conceptual problem in physics.
Following Bridgman in the famous publication, "On the Irreversible Production of Entropy", see in Section 19 Tolman & Fine derive the standard Thomson's (Kelvin) relationships, Eqs., (19.12) and (19.13), connecting the Seebeck, Peltier and Thomson thermo-electric coefficients. Ever since Thomson had derived them in the 1850s and onward his logic or lack of it has been criticized and no book fails to mention this very interesting history. Thomson himself was so dissatisfied with his first attempt that he became convinced of the existence of a third, in addition to Seebeck and Peltier, thermoelectric effect we call now Thomson's heat.
Some 70 years later, nearly contemporaneously, first Bridgman and then Onsager offered two completely different approaches to derive Thomson's equations. Essentially, Bridgman's method is formalized by Tolman & Fine. On the result, here is Zemansky writing in the 5th edition of HEAT AND THERMODYNAMICS(1957)
Attempts to resolve these difficulties were made by Bridgman, by Tolman and Fine, and by Meixner, but the results were not entirely free of objection. The solution is to be found in the macroscopic treatment of irreversible coupled flows developed by Onsager, which was introduced briefly in Chap.9. The following is a simplified version of Onsager's method, based on the work of H. B. Callen.
The same year in "Fashions in Thermodynamics" AJP, 1957 Zemansky also wrote that
The second attempt was made by Tolman and Fine who included the irreversible processes in the thermodynamic treatment in such a way that they cancelled out. Use was made of the concepts of entropy flow and entropy production, but no attempt was made to show how the irreversible effects influenced the reversible ones, if at all. The third and, at present, most acceptable derivation of the Kelvin thermocouple equations is based on Onsager's method of dealing with irreversible flow-processes. In this method, the ideas of entropy flow and entropy production are fundamental.
So Zemansky calls Onsager's method based on the reciprocity relationships more acceptable. In his later books the subject of thermo-electricity is completely left out. Every book I have seen follows Callen's presentation that is based on Onsager's method. Even Bejan, in the very modern "Advanced Engineering Thermodynamics", a book that is otherwise really based on Tolman & Fine, when in Chapter 12 it comes to thermo-electricity it skips Section 19 of T&F completely and goes straight to the reciprocal relationships.
Why? It is the physical nature that lets irreversibility fall out, not Bridgman or Tolman & Fine. Zemansky said that Tolman & Fine, etc., are not entirely free form objections. What are these objections, what specific assumption of Section 19 of Tolman & Fine is unsupported, wrong, etc.?
