In their Comment, Eriksson and Rusanov have explained the steps they took in deriving the Shuttleworth equation. We have followed their explanation, and we believe them when they say that these are the steps they took.
Bottomley, D. J., Makkonen, L., & Kolari, K. (2009). Reply to: "Comment by J.C. Eriksson, A.I. Rusanov on: Incompatibility of the Shuttleworth equation with Hermann's mathematical structure of thermodynamics" [Surf.Sci. 603(2009) 97]. Surface Science, 603(15), 2350-2351. https://doi.org/10.1016/j.susc.2009.05.003