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Therefore, as thermodynamic subsystems, the dividing lines are nonautonomous to a greater extent than the dividing surfaces. 39) where κnj, κgj,and τgj are the normal curvature, the geodesic curvature, and the geodesic torsion of the dividing line at each point along the length of the line [26,27], relative to the jth dividing surface. 41) 12 John Gaydos, et al. 42) κ 2n1 + κ 2g1 = κ 2n 2 + κ 2g 2 = κ 2n 3 + κ 2g 3 = κ 2 . 43) or, for a three-phase system, by Thus, by analogy with the fundamental equations for bulk and surface phases, the general fundamental equation for dividing lines in the density formalism is u (l ) = u (l )  s (l ) ,ρ(jl ) , θ jk , κ nj , κ gj , τ gj  .

78) where K = c1c2 is the Gaussian curvature. 79) where Φ(J,K) denotes a positive, symmetric but not necessarily homogeneous function of the mean and Gaussian curvatures. Simple polynomial examples are: Φ = a + bJ2 – cK, with both constants b and c much less than a [47] and Φ = b(J – Jο)2 + cK [48]. 80) where Δb denotes the Beltrami operator [49]. 81) and was derived by Schadow in 1922 [50]. 79 is lengthy and involves the fourth-order derivatives of the position vector for the surface. Recent mathematical investigations © 2011 by Taylor and Francis Group, LLC 22 John Gaydos, et al.

Therefore, it is desirable to replace c1 and c2 by equivalent curvature related quantities that are invariant. With these considerations in mind, the simplest geometric parameters that possess the desired characteristics are the first (mean) curvature J and the second (Gaussian) curvature K defined by J = c1 + c2 and K = c1c2 . 23) Using J and K as the two scalar differential invariants of the surface, permits one to write the generalized fundamental equation in the energetic density form as u ( a ) = u ( a )  s ( a ) ,ρ(ja ) , J , K  .

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