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77) designed by Poisson in the nineteenth century to characterize the potential energy of a membrane started to appear . 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  and Φ = b(J – Jο)2 + cK . 80) where Δb denotes the Beltrami operator . 81) and was derived by Schadow in 1922 .
110 can be simplified to © 2011 by Taylor and Francis Group, LLC δx = x δz = xc1δz. 112) where c2 is the curvature of the area along side y. 113) which represents the change in area of the planar patch in terms of the mean curvature, the original area and the normal displacement. 114) where δJ can be evaluated as 1 1 1 1 δJ = δ ( c1 + c2 ) = δ + = − 2 + 2 δz = ( c12 + c22 ) δz = ( 2 K − J 2 ) δz. 114 results in δJ = 2KAδz. 106, involving the variation of the total Gaussian curvature δK can be obtained as δK = AδK + KδA = − KJAδz + KJAδz = 0.
66. If these two projection relations are squared and added, then one obtains a “cosine-rule” expression among the various forces at the contact line. The expression contains three angles of contact and is given by ( γ ( j ) )2 + ( γ ( j1 ) )2 + ( γ ( j2 ) )2 + 2 γ ( j ) γ ( j1 ) cos θ jj1 + 2 γ ( j ) γ ( j2 ) cos θ j2 j + 2 γ ( j1 ) γ ( j2 )cosθ j1 j2 = (σκ )2 . 68) It is possible to eliminate two of these three angles in favor of the cusp angle θj1j2 by performing a rotation of the horizontal and vertical axes.