Margules' Power Series Formula: Deriving Coefficients

[jinayb]

Margules suggested a power series formula for expressing the activity composition variation of a binary system.
lnγ1=α1x2+(1/2)α2x2^2+(1/3)α3x2^3+...
lnγ2=β1x1+(1/2)β2x1^2+(1/3)β3x1^3+...
Applying the Gibbs-Duhem equation with ignoring coefficients αi's and βi's higher than i=3, we can obtain α1=β1=0, β2=α2+α3, β3=-α3

I don't know how that relationship between coefficients is derived.
Also, what would be the relationship when higher than i=4 is ignored?
Please help!

 

[fresh_42]

Could you elaborate a bit? E.g.

a) "Margules suggested ..." where? Reference?
b) Here is explained how you can type formulas on PF: https://www.physicsforums.com/help/latexhelp/
c) "the activity composition variation of a binary system" means what? Forces? Number system?