Caratheodory's Concept of Temperature is not Carathéodory's theorem. I have tried,but I found nothing about this question by searching online. This is what I have seen in a thermodynamics textbook; its author passed away many years ago:
There are three homogeneous systems, and their parameters of the equation of state:
$O=f(x_1,...,x_n)$
$P=g(y_1,...,y_m)$
$Q=h(z_1,...,z_k)$
There is a function $F$ so that $F(x_1,...,x_n;y_1,...,y_m)=0$ when $O=P$. Similarly there are function $G$ and $H$ so that,if $P=Q$ and $O=Q$,then:
$G(x_1,\dots,x_n;z_1,\dots,z_k)=0$
$H(y_1,\dots,y_n;z_1,\dots,z_k)=0$
Suppose that $\frac{∂F}{∂x_1}≠0$ and $\frac{∂G}{∂x_1}≠0$. Then from $F(x_1,\dots,x_n;y_1,\dots,y_m)=0$ and $G(y_1,...,y_n;z_1,...,z_k)=0$ we can find $x_1$:
$x_1=F_1(y_1,...,y_m;x_2,...,x_n)=0$
$x_1=G_1(z_1,...,z_k;x_2,...,x_n)=0$
$F_1=G_1$
In this case,$H(y_1,...,y_m;z_1,...,z_k)=0$ if and only if $F_1$ and $G_1$ have the form:
$F_1=a(x_2,...,x_n)s(y_1,...,y_m)+b(x_2,...,x_n)$
$G_1=a(x_2,...,x_n)r(z_1,...,z_k)+b(x_2,...,x_n)$
Let $f(x_1,...,x_n)=\frac{x_1-b(x_2,...,x_n)}{a(x_2,...,x_n)}$,then:
$f(x_1,...,x_n)=s(y_1,...,y_m)=r(z_1,...,z_k)$.
So I have some questions about this discussion:
①why $x_1=F_1(y_1,...,y_m;x_2,...,x_n)=G_1(z_1,...,z_k;x_2,...,x_n)=0$,when $\frac{∂F}{x_1}≠0$ and $\frac{∂G}{x_1}$≠0?
②why $F_1$ and $G_1$ have to be that form in the discussion?
③why we can let $f(x_1,...,x_n)=\frac{x_1-b(x_2,...,x_n)}{a(x_2,...,x_n)}$,why we have to do this?
④Is $g(y_1,...y_m)=s(y_1,...,y_m)$ and $h(z_1,...z_k)=r(z_1,...,z_k)$,in other words,is $s(y_1,...,y_m)$ and $r(z_1,...,z_k)$ are the equation of state of $P$ and $Q$?
I have post this question on mathematics section (https://math.stackexchange.com/questions/3173140/questions-about-using-mathematical-methods-to-prove-the-caratheodorys-concept-o).