I've just started reading Chandrasekhar's Introduction to the Study of Stellar Structure, and I'm having trouble following one of his mathematical assertions. Rather than quote the relevant parts in their entirety here, I've typeset them and linked them here. (For those interested, the entire book is available from the Internet Archive). I hope using an outside link isn't bad manners in forums. What I don't understand, and would like someone to explain, is why the transitivity of thermal equilibrium is both sufficient and necessary (cf "this is then, and only then, possible...") for the condition of thermal equilibrium to have the form t1(p1,V1) - t2(p2,V2) = 0 (same as Eq (4) in Chandrasekhar, but where I've used subscripts instead of bars). Clearly anything of that form implies transitivity, but I don't understand why transitivity implies that form. Any help?