## Abstract

Recently, it has been established that a well-known system of classical shell theory descriptive of membranes in equilibrium is, in fact, amenable to the techniques of soliton theory. Here, it is demonstrated that its canonical discrete counterpart governing ‘plated’ membranes in equilibrium is likewise integrable in that it admits both a parameter-dependent linear representation (Lax pair) and a Bäcklund transformation.

## 1. Introduction

The theory of integrable systems (soliton theory) is multifaceted and extends to a variety of areas in mathematics and physics. Amongst others, there have been two important recent developments which are both geometric in nature but which have otherwise unfolded independently. On the one hand, it has become evident that the area of ‘discrete differential geometry’, which seeks to identify and make use of canonical discrete analogues of differential geometric objects, provides key new insight into the origins of integrable systems and their integrability-preserving discretizations (see the monograph, Bobenko & Seiler 1999 and references therein). On the other hand, until recently, the nonlinear equations descriptive of solitonic behaviour in physical systems had been derived by approximation or expansion methods (except for the Ernst equation of general relativity; Hoenselaers & Dietz 1984). However, it turns out that, remarkably, there exists ‘exact’ hidden integrable structure in diverse areas of nonlinear continuum mechanics such as hydrodynamics, magnetohydrodynamics, the kinematics of fibre-reinforced materials and elastostatics of shell membranes (see the review article by Rogers & Schief 2003*b* and references therein).

In this paper, the above-mentioned two strands are brought together for the first time. Thus, we investigate in detail a discrete model of (shell) membranes in equilibrium together with their resultant internal stress distributions in the absence of external forces. The discrete membranes are composed of planar quadrilateral elements (‘plates’) which are not entirely arbitrary but may be inscribed in circles. The latter property is motivated by the fact that in discrete differential geometry (Bobenko & Seiler 1999) and computer-aided surface design (Gregory 1986) quadrilaterals inscribed in circles have been identified as canonical discrete analogues of surface ‘patches’ which are bounded by pairs of lines of curvature. In mathematical terms, the mid-surfaces of the ‘plated’ membranes therefore constitute standard discrete curvature lattices. We derive a set of equilibrium equations which reduces in the natural continuum limit to that associated with classical membranes (Novozhilov 1964). We then show that, in the case of vanishing ‘shear stresses’, the equilibrium equations admit a parameter-dependent linear representation (Lax pair; Ablowitz & Segur 1981) which may be used to construct explicitly large classes of plated membranes in equilibrium via an associated Bäcklund transformation (Rogers & Schief 2002). This result constitutes the discrete analogue of the recent observation that classical membranes on which the principal lines of stress coincide with the lines of curvature are integrable (Rogers & Schief 2003*a*) and highlights the validity of the standard discrete model of lines of curvature.

## 2. Classical shell membrane theory

In the classical theory of thin shells (Love 1927; Novozhilov 1964), it is customary to replace the three-dimensional stress tensor *σ*_{ik} of elasticity theory defined throughout the shell by statically equivalent internal forces and moments acting on the mid-surface *Σ* of the shell. If no external forces and moments are present (except on the boundary) then the shell is in equilibrium if both the total force and total moment acting on any infinitesimal shell (mid-surface) element vanish. The vanishing of the total force translates into the differential equations (Dikmen 1982)(2.1)where the tensor and vector encode the internal forces. Here, semicolons denote covariant derivatives, indices are raised and lowered by means of the metric coefficients *g*_{ab}, and *h*_{ab} represents the second fundamental form so that(2.2)with the local coordinates *x*^{a} parametrising the mid-surface *Σ*. The requirement of vanishing total moment may be shown to lead to(2.3)where the tensor encapsulates the internal moments and *q*_{[ab]}=(*q*_{ab}−*q*_{ba})/2 constitutes the anti-symmetric part of any tensor *q*_{ab}. In general, in order to determine the stress distribution for any given shape of the membrane, the under-determined set of equilibrium equations (2.1) and (2.3) must be supplemented by constitutive equations which relate the stress tensor _{ik} to the strain tensor _{ik} (Novozhilov 1964). However, in the case of shell membranes, which are defined by the absence of internal moments , it is sufficient to deal with the equilibrium equations(2.4)which are now seen to be well-determined if appropriate boundary conditions are imposed. In the present paper, we are concerned with (discrete) shell membranes which, for brevity, are identified with their mid-surfaces *Σ*.

In the following, we choose to parametrize the membrane in terms of curvature coordinates (*x*, *y*) so that the fundamental forms are given by(2.5)where *κ*_{1} and *κ*_{2} designate the principal curvatures of the membrane. The coefficients of the fundamental forms are constrained by the Gauß–Mainardi–Codazzi equations (Eisenhart 1960)(2.6)In terms of this particular coordinate system, the resultant internal stress components are related to the tensors and by(2.7)so that the equilibrium equations (2.4) reduce to(2.8)and the symmetry condition *T*_{12}=*T*_{21}=*S*.

For any given membrane geometry and associated fundamental forms, the equilibrium equations (2.8) constitute a well-determined *linear* system of differential equations for the resultant stress components *T*_{1}, *T*_{2} and *S*. However, if one assumes that the lines of principal stress coincide with the lines of curvature so that *S*=0, then the equilibrium and Gauß–Mainardi–Codazzi equations become a coupled *nonlinear* system of differential equations. This implies that the corresponding class of membrane geometries is privileged. In fact, these membrane geometries have recently been located by Rogers & Schief (2003*a*) in a large class of integrable surfaces, the so-called O surfaces (Schief & Konopelchenko 2003). Thus, it turns out that the classical equilibrium equations of shell membrane theory subject to vanishing resultant shear stress *S*=0 are amenable to the techniques of soliton theory (Ablowitz & Segur 1981). It is the aim of the present paper to demonstrate that, remarkably, the equilibrium equations associated with an analogous class of plated membranes are likewise integrable. Moreover, it is shown that, as in the classical case, any Combescure transform (Eisenhart 1960) of a membrane may be interpreted as another membrane with the same stress distribution. In the classical case, this implies that, locally, any membrane constitutes a Combescure transform of a minimal surface. In this connection, it is noted that the equilibrium equations (2.8) reduce to the geometric condition of vanishing mean curvature(2.9)if the resultant stress components are taken to be *S*=0 and *T*_{1}=*T*_{2}=const. This confirms the well-known fact that minimal surfaces may be interpreted as shell membranes with associated ‘homogeneous’ resultant stress distribution.

In view of the ‘discrete’ membrane model to be proposed in the §3, it is enlightening to recall briefly the standard derivation (Novozhilov 1964) of the equilibrium equations (2.8). Thus, we consider an infinitesimal membrane element d*Σ* bounded by pairs of lines of curvature. If **F**_{1} and **F**_{2} denote the resultant internal stresses acting on infinitesimal cross-sections *x*=const. and *y*=const. of the membrane, respectively, then the forces acting on d*Σ* are given by −**F**_{1}, −**F**_{2} and **F**_{1}+d**F**_{1}, **F**_{2}+d**F**_{2} as indicated in figure 1. If the membrane is in equilibrium then the total force acting on d*Σ* must vanish and hence(2.10)Moreover, if * r* designates the position vector of the membrane and we introduce the differentials , then the vanishing of the total moment (with respect to any point) acting on d

*Σ*may be expressed as(2.11)In order to evaluate the above pair of conditions, it is noted that the Gauß–Mainardi–Codazzi equations (2.6) constitute the compatibility conditions for the Gauß–Weingarten equations (Eisenhart 1960)(2.12)with(2.13)Here, the orthogonal vectors

*,*

**X***and*

**Y***=*

**N***×*

**X***are the unit tangent vectors to the lines of curvature and the unit normal to the membrane, respectively. The position vector*

**Y***is obtained by integration of the compatible pair(2.14)Decomposition of*

**r**

**F**_{1}and

**F**_{2}into resultant stress components per unit length according to(2.15)reduces the equilibrium condition (2.11) to(2.16)so that

*N*

_{1}=

*N*

_{2}=0 and

*T*

_{12}=

*T*

_{21}=

*S*as required. Substitution of these values into (2.10) then leads to the equilibrium equations (2.8).

## 3. Plated shell membranes

In the area of ‘integrable discrete differential geometry’ (Bobenko & Seiler 1999) and, indeed, in computer-aided surface design (Gregory 1986), the canonical discrete analogue of a ‘small’ patch of a surface bounded by two pairs of lines of curvature turns out to be a planar quadrilateral which is inscribed in a circle. It is therefore natural to consider a discrete (shell) membrane which is composed of plates, each of which may be inscribed in a circle. Accordingly, the corresponding mid-surface may be interpreted as a quadrilateral lattice *Σ* with the quadrilaterals representing the plates. If we confine ourselves to lattices of combinatorics, then the vertices of the lattice are the images of a map(3.1)Thus, the corners of any quadrilateral membrane element (plate) δ*Σ* may be denoted by (cf. figure 2)(3.2)In the following, the arguments of any function depending on the discrete independent variables *n*_{1} and *n*_{2} are suppressed and unit increments are represented by bracketed subscripts as indicated above.

### (a) Internal forces and equilibrium conditions

We now propose the following model for the internal forces acting on a plated membrane: for any quadrilateral, there exist four forces acting on the mid-points of the four corresponding edges and the two forces acting on the common edge of two adjacent quadrilaterals cancel each other. Thus, if we introduce two ‘force fields’, **F**_{1} and **F**_{2}, defined on the edges [* r*,

**r**_{(2)}] and [

*,*

**r**

**r**_{(1)}], respectively, then the forces acting on a quadrilateral membrane element δ

*Σ*are given by −

**F**_{1}, −

**F**_{2}and

**F**_{1(1)},

**F**_{2(2)}(cf. figure 2). This model represents the assumption that the internal stresses are concentrated at the interfaces between plates so that the force fields

**F**_{1}and

**F**_{2}may be interpreted as the resultant internal stresses provided that these are evenly distributed along corresponding edges of the mid-surface. It is evident that for small quadrilaterals, plated membranes may be regarded as (formal) approximations of smooth membranes.

A plated membrane is in equilibrium if the total force and the total moment acting on any plate vanish. The first condition is evidently represented by(3.3)while the second condition adopts the form(3.4)Combination of these relations produces(3.5)which confirms that (3.4) is independent of the point with respect to which the moments are defined. It is readily seen that the pair of equilibrium conditions (3.3) and (3.5) reduces to the pair (2.10) and (2.11) in the formal limit of infinitesimal quadrilaterals.

### (b) The geometric equations

In order to evaluate the above equilibrium conditions, it is necessary to formulate algebraically the planarity and cyclicity of the plates. Thus, if we adopt the decomposition(3.6)where * X*,

*denote unit ‘tangent vectors’ then the planarity of the quadrilaterals implies that the tangent vectors*

**Y**

**X**_{(2)}and

**Y**_{(1)}may be expressed as linear combinations of

*and*

**X***. In the case of quadrilaterals inscribed in circles, these have been shown to be given by (Konopelchenko & Schief 1998; Schief 2003)(3.7)where the edge lengths , and , are related to the coefficients*

**Y***p*and

*q*by(3.8)It is noted that the conditions

**X**^{2}=

**Y**^{2}=1 imply and are, in fact, equivalent (modulo an appropriate normalization) to either of the two relations(3.9)Moreover, the compatibility condition

**r**_{(12)}=

**r**_{(21)}associated with the pair (3.6) is indeed satisfied modulo (3.7) and (3.8). Finally, the linear system (3.7) gives rise to the relation(3.10)which together with (3.9)

_{1}imply that the orientation of the tangent vectors is such that the classical cross-ratio

*Q*(

*,*

**r**

**r**_{(1)},

**r**_{(12)},

**r**_{(2)}) of the co-planar vertices of any quadrilateral regarded as points on the complex plane is given by(3.11)Since a planar quadrilateral has non-intersecting edges if and only if the cross-ratio is negative, we demand that(3.12)

### (c) The equilibrium equations

As in the classical case, the class of geometries admitted by plated membranes is restricted if no resultant ‘shear’ stresses are present, that is if the force fields **F**_{1} and **F**_{2} are orthogonal to the respective edges so that(3.13)for some vector-valued functions * U* and

*. In fact, we now show that, once again, the equilibrium equations become nonlinear due to their coupling with the geometric ‘discrete Gauß equations’ (3.7). Thus, insertion of*

**V**

**F**_{1}and

**F**_{2}into the equilibrium condition (3.3) produces(3.14)Accordingly, the expressions in the square brackets must be linear combinations of

*and*

**X***. On introduction of three functions of separation*

**Y***μ*,

*ν*and

*ρ*, we conclude that(3.15)However, consistency requires that

**U**_{(2)}.

**X**_{(2)}=

**V**_{(1)}.

**Y**_{(1)}=0, whence combination of the relations (3.7) and (3.15) and subsequent evaluation deliver the constraints

*μ*−

*pρ*=

*ν*+

*qρ*and(3.16)where

*φ*=

*μ*−

*pρ*and

*ψ*=

*ρ*(1−

*pq*) so that(3.17)A straightforward but somewhat tedious calculation shows that the second equilibrium condition (3.5) adopts the form(3.18)Since the terms proportional to

*and*

**U***vanish, we obtain only two additional equilibrium equations. These may be brought into the compact form(3.19)which together with the condition (3.16) constitute three linear homogeneous equations for the quantities(3.20)The corresponding determinant is proportional to and non-zero by virtue of the embeddedness requirement (3.12). Thus, the quantities (3.20) must vanish and we may state the following key theorem.*

**V***The equilibrium equations for plated membranes subject to vanishing resultant shear stresses are given by*(3.21)together with(3.22)*Any solution of this system determines an infinite number of Combescure-related membranes in equilibrium with the same resultant stress distribution*(3.23)*via* *‘**integration**’* *of the linear system*(3.24)

It is evident that any solution of the equilibrium equations (3.21) and (3.22) determines only the tangent vectors * X*,

*. Different solutions of the linear system (3.24)*

**Y**_{2,4}give rise to plated membranes whose corresponding edges are parallel but differ in length. In discrete differential geometry, a mapping between lattices consisting of planar quadrilaterals which preserves tangent vectors is termed a discrete Combescure transformation (Konopelchenko & Schief 1998; Schief 2003). As mentioned earlier, the relevance of the Combescure transformation in classical membrane theory has been discussed in Rogers & Schief (2003

*a*).

## 4. Integrability: a Bäcklund transformation and a Lax pair for plated membranes

We now demonstrate that the equilibrium equations derived in §3*c* are integrable in the sense of soliton theory. Thus, we derive a Bäcklund transformation (Rogers & Schief 2002) and a Lax pair (Ablowitz & Segur 1981) which may, in principle, be used to construct iteratively large classes of plated membranes in equilibrium. We adopt an approach which has recently been investigated in detail in connection with the isolation and classification of discrete integrable systems (Nijhoff & Walker 2001; Bobenko & Suris 2002; Nijhoff 2002; Adler *et al*. 2003). To this end, it is observed that the equilibrium equations (3.21) and (3.22) may be regarded as relations between objects which are ‘attached’ to any given quadrilateral (plate). It is therefore natural to inquire as to whether it is *consistent* to demand that these relations hold on every quadrilateral of a three-dimensional lattice of combinatorics. This amounts to determining whether the associated compatibility conditions are satisfied.

### (a) Three-dimensional consistency

In order to make use of the above-mentioned ‘consistency approach’, it is necessary to introduce a pair of arbitrary ‘parameters’ whose natural counterpart may then be interpreted as a Bäcklund parameter. The latter constitutes the key ingredient in the iterative application of Bäcklund transformations. It turns out that such parameters are naturally injected into the equilibrium equations by relaxing the normalization * U*.

*=*

**X***.*

**V***=0 and considering the transition(4.1)where*

**Y***α*and

*β*are functions of their indicated arguments. In index notation, the equilibrium equations (3.21) then become(4.2)for

*i*≠

*k*∈{1,2}, with the identifications(4.3)and the ‘frame’ equations (3.22) for the tangent vectors read(4.4)The systems (4.2) and (4.4) are defined on the square lattice . If we now assume that the quantities involved depend on an additional independent variable

*n*

_{3}, then this system may be extended to the cubic lattice , if appropriate dependent variables carrying an index 3 are introduced and

*i*≠

*k*∈{1, 2, 3}. However, it is now required to analyse any resultant compatibility conditions. Thus, the compatibility conditions

**X**_{i(kl)}=

**X**_{i(lk)},

*i*≠

*k*≠

*l*≠

*i*associated with the extended frame equations (4.4) give rise to the nonlinear system(4.5)for the six coefficients

*p*

_{ik}. This well-known integrable system (Bogdanov & Konopelchenko 1995) is of a purely geometric nature and, in fact, guarantees that the unit vectors

**X**_{i}may be regarded as tangent vectors to the three-dimensional lattices(4.6)whose quadrilaterals are planar. Lattices of this type constitute discrete analogues of classical conjugate coordinate systems in (Doliwa & Santini 1997; Bobenko & Seiler 1999). These are obtained by integration of(4.7)which represents the extended version of the linear system (3.24). Here, it is noted that (4.7)

_{1}is indeed compatible modulo the systems (4.4)

_{1}and (4.7)

_{2}. Moreover, the conditions are known to constitute admissible constraints which ensure that the quadrilaterals of the conjugate lattices are inscribed in circles (Konopelchenko & Schief 1998). Accordingly, the mappings

*determine discrete analogues of classical orthogonal coordinate systems in (Bobenko 1999; Bobenko & Seiler 1999).*

**r**The consistency of the equilibrium equations (4.2) is guaranteed if the compatibility condition **U**_{i(kl)}=**U**_{i(lk)} is satisfied and the constraint *ψ*_{ik}+*ψ*_{ki}=0 is preserved in the *n*_{l} -direction. The latter is readily verified by means of the relation(4.8)which implies that *ψ*_{ik(l)}+*ψ*_{ki(l)}=0. Secondly, evaluation of the quantity *Γ*_{ik(l )} reveals that the expression *Γ*_{ik(l)}*Γ*_{il}*Γ*_{lk} is symmetric in *i*, *k*, *l*. Accordingly, since the quantity *Γ*_{ik(l)}*Γ*_{il}**U**_{i(kl)} may be shown to be symmetric in *k*, *l*, the vector **U**_{i(kl)} admits the same symmetry. Hence, the remaining compatibility condition is satisfied and the following theorem obtains.

*Any solution of the* *(three-dimensional )* *compatible systems* *(4.2), (4.4)* *and (4.5)* *with the normalisation α*_{1}=*α*_{2}=0 *may be interpreted as a one-parameter* *(n*_{3}*)* *family of solutions of the equilibrium equations* *(3.21)* *and* *(3.22)*. *The associated plated membranes in equilibrium are obtained by integration of the compatible linear system* *(4.7)*.

In the remaining two sections, it is shown that the transition from the discrete membrane to the discrete membrane for some integer *n* _{0} may be interpreted as a standard Bäcklund transformation. In particular, it is demonstrated that the Bäcklund transformation may be applied to *any* seed discrete membrane and the construction of its Bäcklund transform involves the solution of a system of *linear* discrete equations.

### (b) A Lax pair

We now investigate in detail the relationship between two plated membranes *Σ*=*Σ*(*n*_{3}) and *Σ*′=*Σ*(*n*_{3}+1) for a fixed integer *n*_{3} as encapsulated in theorem 4.1. To this end, it proves convenient to set down explicitly the extended system (4.2), (4.4) and (4.5). We are concerned with the original normalisation of the vectors * U* and

*corresponding to*

**V***α*=

*β*=0, so that the equations which do not carry an index 3 are displayed in theorem 3.1. The equations which involve a unit increment of

*n*

_{3}(indicated by a prime) are given by(4.9)These provide the transition from the solution of the equilibrium equations associated with the discrete membrane

*Σ*to that corresponding to

*Σ*′ and involve the functions(4.10)which obey the nonlinear system(4.11)In the following, it is demonstrated that, remarkably, the above system may be

*linearized*. Thus, we begin with the introduction of a scalar function

*M*according to(4.12)and consider the change of variables(4.13)The corresponding compatibility condition

*M*

_{(12)}=

*M*

_{(21)}is readily shown to be satisfied modulo the system (4.11)

_{5–8}which, in terms of the new variables, reads(4.14)It is seen that the pairs (4.14)

_{1,2}and (4.14)

_{3,4}exhibit the same structure as (3.22)

_{1,3}and (3.24)

_{2,4}, respectively. In fact, in the terminology of soliton theory, the quantities

*,*

**X***X*and ,

*H*represent eigenfunctions and adjoint eigenfunctions, respectively. Moreover, insertion of

*as given by (4.13)*

**Z**_{1}into the pair (4.11)

_{1,2}produces(4.15)which represents a vector version of the defining equations (4.12) written in the form(4.16)Thus,

*and*

**M***M*constitute so-called ‘squared eigenfunctions’. It is noted that the importance of squared eigenfunctions in the construction of Bäcklund transformations has been well documented (see Rogers & Schief 2002 and references therein). Furthermore, the constraint

**Z**^{ 2}=1 yields(4.17)which, inserted into (4.16), provides the relationships(4.18)On use of (4.15), these may be solved for the adjoint eigenfunctions

*H*and

*K*. One obtains the expressions(4.19)which constitute particular solutions of (4.14)

_{3,4}as is required for consistency.

The additional change of variables(4.20)now transforms the pair (4.11)_{3,4} into(4.21)and the relations (4.11)_{11,12} deliver(4.22)Combination with (4.11)_{9,10} produces the expressions(4.23)for the eigenfunctions *X* and *Y* with *λ*=1/*μ*. Once again, these are consistent with (4.14)_{1,2}. Finally, the remaining equation (4.11)_{13} gives rise to(4.24)In summary, it emerges that the systems (4.15) and (4.21) are *linear* in the vector-valued functions * M* and by virtue of the expressions (4.19) and (4.23) for the (adjoint) eigenfunctions

*H*,

*K*and

*X*,

*Y*, respectively. The quadratic constraint (4.24) turns out to be an associated particular first integral. In fact, setting aside the genesis of this linear system, one may formulate this result as follows.

*The linear system*(4.25)*where the matrices A and B are given by*(4.26)*is compatible modulo the equilibrium equations* (3.21) *and* (3.22) *for plated membranes*. *It admits the first integral*(4.27)*Here,* *λ constitutes an arbitrary constant parameter*.

In §4*c*, it is shown that the linear systems (4.25) and (4.26) constitute a proper ‘linear representation’ or Lax pair (Ablowitz & Segur 1981) in that it may be used to define a Bäcklund transformation for plated membranes, wherein *λ* plays the role of the Bäcklund parameter. In fact, this system together with its quadratic first integral (4.27) are reminiscent of the Lax pairs and first integrals recorded in the context of the integrable class of discrete O surfaces (Schief 2003). Whether there exists a connection between the geometry of plated membranes and discrete O surfaces is currently being investigated. As mentioned earlier, in the classical case, such a connection has been established in Rogers & Schief (2003*a*).

### (c) A Bäcklund transformation

It is evident that the mapping (4.9) between the solutions of the equilibrium equations associated with the plated membranes *Σ* and *Σ*′ may be expressed entirely in terms of quantities related to *Σ* and the functions * M*, . In addition, the relation between the position vectors of

*Σ*and

*Σ*′ is given by (4.7) for

*i*=3. If we set (and , ) then we obtain(4.28)Moreover, the lengths of the edges transform according to (4.7)

_{2}for

*k*=3, namely(4.29)In particular, in terms of the function defined by(4.30)the relations (4.28)

_{2,3}become(4.31)so that is seen to be another squared eigenfunction associated with the (adjoint) eigenfunctions , and

*X*,

*Y*.

The transformation laws obtained in the preceding may be summarized as follows.

*Let Σ be a plated membrane associated with a solution of the equilibrium equations* (3.21) *and* (3.22) *and M,*

*be a solution of the linear representation*(4.25)

*and*(4.26)

*subject to the admissible constraint*(4.24).

*Let the quantities H,*

*K,*

*X,*

*Y,*

*, M,*

*be defined by*(4.32)

*Then,*

*the position vector of another plated membrane Σ*′

*is given by*(4.33)

*together with the related geometric quantities*(4.34)

*The corresponding resultant stress distribution*,

**F**_{2}=

*′×*

**U***′*

**X***is obtained from*(4.35)

In geometric terms, the above transformation between the two plated membranes represents a Bäcklund transformation which constitutes nothing but a particular case of the standard discrete analogue (Schief & Konopelchenko 2003; Schief 2003) of the classical Ribaucour transformation (Eisenhart 1962) which preserves lines of curvature on surfaces. This is due to the fact that, by construction (cf. §4*a*), the six quadrilaterals [* r*,

**r**_{(1)},

**r**_{(12)},

**r**_{(2)}], , , , and are planar and, in fact, inscribed in circles (Bobenko 1999).

## 5. Perspectives

It has been shown that the equilibrium equations associated with plated membranes subject to vanishing resultant shear stresses and external forces are amenable to the techniques of soliton theory. In the classical case of shell membranes, this observation has been made only recently. In fact, in Rogers & Schief (2003*a*), it has been demonstrated that integrability is still present if the membranes are exposed to constant normal pressure. Since the latter situation is of particular physical importance, it is now necessary to investigate whether an analogous result may be found in the case of plated membranes.

Another topic of current interest is the establishment of rigorous theorems which state under what circumstances and in which sense continuous integrable geometries are approximated by their canonical discrete counterparts. In the case of discrete analogues of orthogonal coordinate systems, such theorems have been recorded in Bobenko *et al*. (2003). Since the significance of these coordinate systems in the current context has been established in §4*a*, one may now be able to prove that the discrete membranes considered here may indeed be used as approximations of smooth membranes. This may be relevant, in particular, to the numerical integration of the classical equilibrium equations. It is noted that finite element modelling of plates and shells based on ‘discrete Kirchhoff techniques’ (Batoz *et al*. 2001) has been a subject of extensive research.

## Footnotes

- Received February 3, 2005.
- Accepted May 31, 2005.

- © 2005 The Royal Society