How can the simplification of equation I.4 using equation I.5 be justified?

[Aniket1]

I was studying zeroth law from MIT Courseware (http://ocw.mit.edu/courses/physics/...-fall-2013/lecture-notes/MIT8_333F13_Lec1.pdf). On page 2, it is mentioned that equation I.4 can be simplified using equation I.5 by cancelling the co-ordinates of C. Could someone guide me justify this fact? I tried working it out by assuming functions and I could construct functions where equation I.4 and equation I.5 are satisfied but it is not possible to cancel C. Am I missing something?

 

[Useful nucleus]

Equation I.4 does not contain C1. Now let' eliminate C2. Differentiate I.4 with respect to A1, then you will get

F^#_AC(A1,A2,...,C2,...)=0

But then you can write this equation as C2=G_AC(A1,A2,...,C3,C4,...).

Similarly, differentiate I.4 with respect to B1. Then you will get
F^#_BC(B1,B2,...,C2,...)=0

But again you can write it as C2=G_BC(B1,B2,...,C3,C4,...).

Since there is equilibrium then,
G_BC(B1,B2,...,C3,C4,...) = G_AC(A1,A2,...,C3,C4,...).

Thus, you eliminated C2. Repeat the procedure unti you eliminate all C's . Then you will end up with
θ_A(A1,A2,...)=θ_B(B1,B2,...).

 

[Useful nucleus]

Also note that MIT recorded professor Kardar's lectures two years ago, so you can watch him explaining his notes:

http://ocw.mit.edu/courses/physics/...hanics-of-particles-fall-2013/video-lectures/

 

[Aniket1]

Also note that MIT recorded professor Kardar's lectures two years ago, so you can watch him explaining his notes:

http://ocw.mit.edu/courses/physics/...hanics-of-particles-fall-2013/video-lectures/

Thanks a lot!