MATEMATIQKI VESNIK 49 (1997), 51–58

UDK 514.752.4

originalni nauqni rad research paper

ON THE SECOND ORDER INFINITESIMAL BENDINGS OF A CLASS OF TOROIDS

Ljubica S. Velimirović

Abstract. This paper is devoted to a study of the second order infinitesimal bendings of a class of toroids generated by a simple polygonal meridian. A necessary and sufficient condition for the existence of the second order infinitesimal bendings is determined.

1. Introduction

Let us suppose that the meridian of a rotational surface T is a simple polygon with apeces A_{i}(u_{i}, ρ_{i}), ρ_{i} > 0, i = 1,…, n, A_{n+1} ≡ A_{1} at the orthogonal coordinate system uOρ, with orths ē, ā(v), ā′(v), where ē is unit vector of the axis of rotation, ā(v) unit vector of the ρ-axis where v is the angle between the plane of initial position of the meridian and ā(v), then ā′(v) ⊥ ā(v) and ā′(v) ⊥ ē (see [1], page 90). The equations of the sides are

ρ_{(i)} = ρ_{i} + \frac{ρ_{i+1} − ρ_{i}}{u_{i+1} − u_{i}} (u − u_{i}) = k_{i}u + n_{i},  (1.1)

where ρ_{(i)} is the value of ρ on A_{i}A_{i+1}, and

k_{i} = \frac{ρ_{i+1} − ρ_{i}}{u_{i+1} − u_{i}} , n_{i} = ρ_{i} − \frac{ρ_{i+1} − ρ_{i}}{u_{i+1} − u_{i}} .  (1.2)

By rotation of the mentioned polygon round the u-axis one gets the toroid T. Let us consider the second order infinitesimal bendings of such a toroid.

We shall use here Cohn-Vossen’s method [1], [2] and [3]. Infinitesimal bendings of the first order for the toroid rotational surfaces with polygonal meridian were considered at [4] and [5].

The correspondence between the volume variation and the flow of the infinitesimal bending for piecewise smooth surfaces was studied at remarkable paper [8].

AMS Subject Classification: 53 A 05

Keywords and phrases: Second order infinitesimal bendings, toroid.

Communicated at the 11. Yugoslav Geometrical Seminar, Divčibare 1996.

Supported by Grant 04M03D of RFNS through Math.Inst.SANU.

52 Lj. Velimirović

References for later investigations on infinitesimal bendings can also be found at [6] and [7].

The radius vector of surface T is represented in the known way [1]

r̄(u, v) = uē + ρ(u)ā(v),  (1.3)

where ρ = ρ(u) is the equation of the meridian. Let

z̄^{(1)}(u, v) = α^{(1)}(u, v)ē + β^{(1)}(u, v)ā + γ^{(1)}(u, v)ā′,  (1.4)
z̄^{(2)}(u, v) = α^{(2)}(u, v)ē + β^{(2)}(u, v)ā + γ^{(2)}(u, v)ā′  (1.5)

be continuous fields of infinitesimal bendings of the first and the second order respectively, where z̄^{(2)}(u, v) is an extension of the field z̄^{(1)}(u, v). The fields z̄^{(1)}(u, v) and z̄^{(2)}(u, v) satisfy the systems of differential equations [1]:

α^{(1)}_{u} + ρ′ β^{(1)}_{u} = 0, β^{(1)} + γ^{(1)}_{v} = 0, α^{(1)}_{v} + ρ′( β^{(1)}_{v} − γ^{(1)}) + ρ γ^{(1)}_{u} = 0,  (1.6)
α^{(2)}_{u} + ρ′ β^{(2)}_{u} = −\frac{1}{2} [ α^{(1)}_{u}^{2} + β^{(1)}_{u}^{2} + γ^{(1)}_{u}^{2}] ,
β^{(2)} + γ^{(2)}_{v} = −\frac{1}{2ρ} [ α^{(1)}_{v}^{2} + ( β^{(1)}_{v} − γ^{(1)})^{2}] ,  (1.7)
α^{(2)}_{v} + ρ′( β^{(2)}_{v} − γ^{(2)}) + ρ′ γ^{(2)}_{u} = − [ α^{(1)}_{u} α^{(1)}_{v} + β^{(1)}_{u}( β^{(1)}_{v} − γ^{(1)})] .

According to [1] we have the first order fundamental field

z̄^{(1)}_{k}(u, v) = α^{(1)}_{k}(u, v)ē + β^{(1)}_{k}(u, v)ā + γ^{(1)}_{k}(u, v)ā′, k ≥ 2, k ∈ N  (1.8)

where

α^{(1)}_{k}(u, v) = φ^{(1)}_{k}(u)e^{ikv} + φ^{(1)}_{−k}(u)e^{−ikv},
β^{(1)}_{k}(u, v) = ψ^{(1)}_{k}(u)e^{ikv} + ψ^{(1)}_{−k}(u)e^{−ikv},  (1.9)
γ^{(1)}_{k}(u, v) = χ^{(1)}_{k}(u)e^{ikv} + χ^{(1)}_{−k}(u)e^{−ikv}.

The field of infinitesimal bendings z̄^{(2)}_{2k}(u, v) of the second order is

z̄^{(2)}_{2k}(u, v) = α^{(2)}_{2k}(u, v)ē + β^{(2)}_{2k}(u, v)ā + γ^{(2)}_{2k}(u, v)ā′, k ≥ 2, k ∈ N,  (1.10)

where

\begin{aligned}α^{(2)}_{2k}(u, v) & = φ^{(2)}_{2k}(u)e^{2ikv} + φ^{(2)}_{−2k}(u)e^{−2ikv} + φ^{(2)}_{0}(u), \\ β^{(2)}_{2k}(u, v) & = ψ^{(2)}_{2k}(u)e^{2ikv} + ψ^{(2)}_{−2k}(u)e^{−2ikv} + ψ^{(2)}_{0}(u), \\ γ^{(2)}_{2k}(u, v) & = χ^{(2)}_{2k}(u)e^{2ikv} + χ^{(2)}_{−2k}(u)e^{−2ikv} + χ^{(2)}_{0}(u). \end{aligned}  (1.11)

On the second order infinitesimal bendings 53

The functions φ^{(1)}_{k}(u), ψ^{(1)}_{k}(u), χ^{(1)}_{k}(u), φ^{(2)}_{2k}(u), ψ^{(2)}_{2k}(u), χ^{(2)}_{2k}(u), φ^{(2)}_{0}(u), ψ^{(2)}_{0}(u),

χ^{(2)}_{0}(u) satisfy the following systems of differential equations

φ^{(1)}′_{k} + ρ′ ψ^{(1)}′_{k} = 0, ik χ^{(1)}_{k} + ψ^{(1)}_{k} = 0, ik φ^{(1)}_{k} + ρ′( ψ^{(1)}_{k} − χ^{(1)}_{k}) + ρ χ^{(1)}_{k} = 0, (1.12)
\begin{aligned}φ^{(2)}′_{2k} + ρ′ ψ^{(2)}_{2k} & = −\frac{1}{2} [ φ^{(1)}_{k}^{′2} + ψ^{(1)}_{k}^{′2} + χ^{(1)}_{k}^{′2}] , \\ 2ik χ^{(2)}_{2k} + ψ^{(2)}_{2k} & = −\frac{1}{2ρ} [−k^{2} φ^{(1)}_{k}^{2} + (ik ψ^{(1)}_{k} − χ_{k}^{(1)})^{2}] , \\ 2ik φ^{(2)}_{2k} + ρ′(2ik ψ^{(2)}_{2k} − χ^{(2)}_{2k}) + ρ χ^{(2)}′_{2k} & = −ik φ^{(1)}_{k} φ^{(1)}′_{k} − ψ^{(1)}′_{k}(ik ψ^{(1)}_{k} − χ_{k}^{(1)}), \end{aligned} (1.13)

φ^{(2)}′_{0} + ρ′ ψ^{(2)}′_{0} = − φ^{(1)}′_{k} φ^{(1)}′_{−k} − ψ^{(1)}′_{k} ψ^{(1)}′_{−k} − χ^{(1)}′_{k} χ^{(1)}′_{−k},
ψ^{(2)}_{0} = −\frac{1}{ρ} [k^{2} φ^{(1)}_{k} φ^{(1)}_{−k} − (ik ψ^{(1)}_{k} − χ^{(1)}_{k})(ik ψ^{(1)}_{−k} − χ^{(1)}_{−k})] (1.14)
−ρ′ χ^{(2)}_{0} + ρ χ^{(2)}′_{0} = ik φ^{(1)}′_{k} φ^{(1)}_{−k} − ik φ^{(1)}_{k} φ^{(1)}′_{−k} − ψ^{(1)}′_{k}(−ik ψ^{(1)}_{−k} − χ^{(1)}_{−k})

− ψ^{(1)}′_{−k}(ik ψ^{(1)}_{k} − χ^{(1)}_{k}).

The system (1.13) can be written in the form

\begin{aligned}φ^{(2)}′_{2k} + ρ′ ψ^{(2)}′_{2k} & = A^{(2)}_{2k}(u), \\ 2ik χ^{(2)}_{2k} + ψ^{(2)}_{2k} & = B^{(2)}_{2k}(u), \\ 2ik φ^{(2)}_{2k} + ρ′(2ik ψ^{(2)}_{2k} − χ^{(2)}_{2k}) + ρ χ^{(2)}′_{2k} & = C^{(2)}_{2k}(u), \end{aligned}  (1.13′)

where A^{(2)}_{2k}(u), B^{(2)}_{2k}(u), C^{(2)}_{2k}(u), are the right sides of (1.13). If we express the right sides of (1.13) by ψ^{(1)}_{k} we’ll have according to [2]

φ^{(2)}′_{2k} + ρ′ ψ^{(2)}_{2k} = \frac{1}{2k^{2}} [1 − (1 + ρ^{′2})k^{2}] ψ^{(1)}_{k}^{′2},

2ik χ^{(2)}_{2k} + ψ^{(2)}_{2k} = [(k^{2} − 1)^{2}(1 + ρ^{′2}) ψ^{(1)}_{k}^{2} + 2ρρ′(k^{2} − 1) ψ^{(1)}_{k} ψ^{(1)}′_{k} + ρ^{2} ψ^{(1)}_{k}^{′2}]/(2ρk^{2}),

2ik φ^{(2)}_{2k} + ρ′(2ik ψ^{(2)}_{2k} − χ^{(2)}_{2k}) + ρ χ^{(2)}′_{2k} = −i[(1 + ρ^{′2})(k^{2} − 1) ψ^{(1)}_{k} ψ^{(1)}′_{k} + ρρ′ ψ^{(1)}_{k}^{′2}]/k.

(1.13 ′′)

54 Lj. Velimirović

In the same way, the equations (1.14) are

φ^{(2)}′_{0} + ρ′ ψ^{(2)}_{0} = A^{(2)}_{0}(u),
ψ^{(2)}_{0} = B^{(2)}_{0}(u),  (1.14′)
ρ χ^{(2)}′_{0} − ρ′ χ^{(2)}_{0} = C^{(2)}_{0}(u),

or

φ^{(2)}′_{0} + ρ′ ψ^{(2)}′_{0} = −\frac{1}{k^{2}} [k^{2}(1 + ρ^{′2}) + 1] ψ^{(1)}′_{k} ψ^{(1)}′_{−k},
ψ^{(2)}_{0} = −[(k^{2} − 1)^{2}(1 + ρ^{′2}) ψ^{(1)}_{k} ψ^{(1)}_{−k}

+ ρρ′(k^{2} − 1)( ψ^{(1)}_{−k} ψ^{(1)}′_{k} + ψ^{(1)}′_{−k} ψ^{(1)}_{k}) + ρ^{2} ψ^{(1)}′_{−k} ψ^{(1)}′_{k}]/(ρk^{2}),

ρ χ^{(2)}′_{0} − ρ′ χ^{(2)}_{0} = i(k^{2} − 1)(1 + ρ^{′2})( ψ^{(1)}′_{k} ψ^{(1)}_{−k} − ψ^{(1)}_{k} ψ^{(1)}′_{−k})/k.

(1.14′′)

The system (1.12) is equivalent to the differential equation of the second order with respect to ψ^{(1)}_{k} (see[1]):

ρ ψ^{(1)}′′_{k} + (k^{2} − 1)ρ′′ ψ^{(1)}_{k} = 0,  (1.15)

and the system (1.13) is equivalent to the differential equation (see [2]):

ρ ψ^{(2)}′′_{2k} + (4k^{2} − 1)ρ′′ ψ^{(2)}_{2k} = R^{(2)}_{2k}(u)  (1.16)

where

R^{(2)}_{2k}(u) = −ρ′′ B^{(2)}_{2k}(u) + ρB^{(2)}′′_{2k}(u) − 4k^{2} A^{(2)}_{2k}(u) − 2ik C^{(2)}′_{2k}(u)  (1.17)

i.e.

R^{(2)}_{2k}(u) = \frac{(k^{2} − 1)^{2}ρ^{′2}}{k^{2}ρ^{2}} [1 + ρ^{′2} + ρρ′′(k^{2} − 1) + \frac{k^{2}ρρ′′}{ρ^{′2}} ] ψ^{(1)}_{k}^{2}

− \frac{2ρ′(k^{2} − 1)}{k^{2}ρ}[ρρ′′ + (k^{2} − 1)(1 + ρ^{′2})] ψ^{(1)}_{k} ψ^{(1)}′_{k}

+ \frac{1}{k^{2}} [(1 + ρ^{′2})(1 − k^{2})^{2} − ρρ′′(1 + k^{2})] ψ^{(1)}_{k}^{′2}  (1.17′)

2. Infinitesimal bendings of the second order of a toroid with poligonal meridian

The meridian of our toroid consists of line segments ρ_{(i)}(u) = k_{i}u + n_{i}. From (1.15)

ψ^{(1)}_{k,i}(u) = M^{(1)}_{k,i}u + N^{(1)}_{k,i}, i = 1,…, n, (M^{(1)}_{k,i} = M_{i}, N^{(1)}_{k,i} = N_{i})  (2.1)

i.e. we have the linearity of the field of infinitesimal bendings of the first order, where M_{i}, N_{i} are the constants defined in [4]. For the determination of the second order field of infinitesimal bendings, according to (1.16) we have the equation

ρ_{(i)}(u) ψ^{(2)}′′_{2k,i}(u) = R^{(2)}_{2k,i}(u)  (2.2)

where

R^{(2)}_{2k,i}(u) = \frac{(k^{2} − 1)^{2}ρ_{(i)}^{′2}}{k^{2}ρ_{(i)}^{2}} [1 + ρ_{(i)}^{′2} + ρ_{(i)}ρ′′_{(i)}(k^{2} − 1) + \frac{k^{2}ρ_{(i)}ρ′′_{(i)}}{ρ_{(i)}^{′2}} ] ψ^{(1)}_{k,i}^{2}

− \frac{2ρ′_{(i)}(k^{2} − 1)}{k^{2}ρ_{(i)}}[ρ_{(i)}ρ′′_{(i)} + (k^{2} − 1)(1 + ρ_{(i)}^{′2})] ψ^{(1)}_{k,i} ψ^{(1)}′_{k,i}

+ \frac{1}{k^{2}} [(1 + ρ_{(i)}^{′2})(1 − k^{2})^{2} − ρ_{(i)}ρ′′_{(i)}(1 + k^{2})] ψ^{(1)}_{k,i}^{′2}.  (2.3)
Since ρ′_{(i)} = k_{i}, ρ′′_{(i)} = 0, we have
R^{(2)}_{2k,i}(u) = \frac{(k^{2} − 1)^{2}(1 + k_{i}^{2})}{k^{2}}[\frac{k_{i}^{2}(M_{i}u + N_{i})^{2}}{(k_{i}u + n_{i})^{2}} − 2k_{i}M_{i} \frac{M_{i}u + N_{i}}{k_{i}u + n_{i}} + M_{i}^{2}]. (2.4)

In order to find ψ^{(2)}_{2k,i}(u), we have to solve the differential equation of the second order (2.2), which from (2.3,4) becomes:

ψ^{(2)}′′_{2k,i}(u) = \frac{(k^{2} − 1)^{2}(1 + k_{i}^{2})}{k^{2}}[\frac{k_{i}^{2}(M_{i}u + N_{i})^{2}}{(k_{i}u + n_{i})^{3}} − 2k_{i}M_{i} \frac{M_{i}u + N_{i}}{(k_{i}u + n_{i})^{2}} + \frac{M_{i}^{2}}{(k_{i}u + n_{i})}].

(2.5)

The solution of this equation is

ψ^{(2)}_{2k,i}(u) = \frac{(k^{2} − 1)^{2}(1 + k_{i}^{2})(k_{i}N_{i} − M_{i}n_{i})^{2}}{2k^{2}k_{i}^{2}(k_{i}u + n_{i})} + M^{(2)}_{2k,i}u + N^{(2)}_{2k,i},  (2.6)

where M^{(2)}_{2k,i}, N^{(2)}_{2k,i}, i = 1, 2,…, n are the constants that are to be found. Rotational toroid generated by a polygonal meridian consists of n regular parts. Let us denote them by T_{i}, i = 1, 2,…, n. Let c_{i}, i = 1, 2,…, n, be regular parts of

meridian, c_{i} : ρ_{(i)} = ρ_{(i)}(u). We shall denote by z^{(1)}_{k,i}(u, v) and z^{(2)}_{2k,i}(u, v) the

fields of infinitesimal bendings at T_{i}. The fields z^{(1)}_{k,i}(u, v), z^{(2)}_{2k,i}(u, v) are to be continuous on the whole surface. At the circles circuscribed by the apeces of the meridian, the fields of infinitesimal bendings of the first and the second order are to be continuous [1]. From that fact, at the mentioned points u = σ of the meridian we have for the field of infinitesimal bendings of the first order:

φ^{(1)}_{k,i}(σ) = φ^{(1)}_{k,i+1}(σ), ψ^{(1)}_{k,i}(σ) = ψ^{(1)}_{k,i+1}(σ), χ^{(1)}_{k,i}(σ) = χ^{(1)}_{k,i+1}(σ)  (2.7)

and in the same manner for the field of the second order:

φ^{(2)}_{2k,i}(σ) = φ^{(2)}_{2k,i+1}(σ), ψ^{(2)}_{2k,i}(σ) = ψ^{(2)}_{2k,i+1}(σ), χ^{(2)}_{2k,i}(σ) = χ^{(2)}_{2k,i+1}(σ),(2.8)
φ^{(2)}_{0,i}(σ) = φ^{(2)}_{0,i+1}(σ), ψ^{(2)}_{0,i}(σ) = ψ^{(2)}_{0,i+1}(σ), χ^{(2)}_{0,i}(σ) = χ^{(2)}_{0,i+1}(σ). (2.9)

At the points u = σ of the meridian ρ = ρ(u) we also have the following relations

ρ_{(i)}(σ) = ρ_{(i+1)}(σ), ρ′_{(i)}(σ) ≠ ρ′_{(i+1)}(σ), i = 1, 2,…, n.  (2.10)

According to (1.15,1.16) on the parts T_{i} of the surface T it is

ρ_{(i)} ψ^{(1)}′′_{k,i} + (k^{2} − 1)ρ′′_{(i)} ψ^{(1)}_{k,i} = 0,  (2.11)
ρ_{(i)} ψ^{(2)}′′_{2k,i} + (4k^{2} − 1)ρ′′_{(i)} ψ^{(2)}_{2k,i} = R^{(2)}_{2k,i}, i = 1, 2,…, n.  (2.12)

Based on [1], page 112, for the field of infinitesimal bendings ψ^{(1)}_{k}(u) at the points u = σ we have

\begin{matrix}ρ_{(i)}(σ) & ψ^{(1)}′_{k,i}(σ) + (k^{2} − 1)ρ′_{(i)}(σ) ψ^{(1)}_{k,i}(σ) \\ = ρ_{(i+1)}(σ) ψ^{(1)}′_{k,i+1}(σ) + (k^{2} − 1)ρ′_{(i+1)}(σ) ψ^{(1)}_{k,i+1}(σ) \end{matrix}  (2.13)

For the field of the second order infinitesimal bendings we have according to [2]

\begin{matrix}ρ_{(i+1)}(σ) & ψ^{(2)}′_{2k,i+1}(σ) + (4k^{2} − 1)ρ′_{(i+1)}(σ) ψ^{(2)}_{2k,i+1}(σ) \\ = ρ_{(i)}(σ) ψ^{(2)}′_{2k,i}(σ) + (4k^{2} − 1)ρ′_{(i)}(σ) ψ^{(2)}_{2k,i}(σ) + Q_{k}(σ), \end{matrix}  (2.14)

where

Q_{k}(σ) = \frac{ρ′_{(i)} − ρ′_{(i+1)}}{ρ_{(i)}k^{2}} [(k^{2} − 1)^{2}(ρ_{(i)}^{′2} − k^{2} − ρ′_{(i)}ρ′_{(i+1)}k^{2}) ψ^{(1)}_{k,i}^{2}

+ (1 + k^{2})ρ_{(i)}^{2} ψ^{(1)}_{k,i}^{′2} + ρ_{(i)}(k^{2} − 1)(2ρ′_{(i)} + k^{2}ρ′_{(i)} − k^{2}ρ′_{(i+1)}) ψ^{(1)}_{k,i} ψ^{(1)}′_{k,i}].

At the line generated by A_{i}(u_{i}, ρ_{i}), i = 1, 2,…, n, based on (2.8) we have

\begin{aligned}u_{i} & M^{(2)}_{2k,i} − u_{i}M^{(2)}_{2k,i+1} + N^{(2)}_{2k,i} − N^{(2)}_{2k,i+1} \\ & = \frac{(k^{2} − 1)}{2k^{2}ρ_{i}}[\frac{(1 + k_{i+1}^{2})(k_{i+1}N^{(1)}_{k,i+1} − M^{(1)}_{k,i+1}n_{i+1})^{2}}{k_{i+1}^{2}} \\ & − \frac{(1 + k_{i}^{2})(k_{i}N^{(1)}_{k,i} − M^{(1)}_{k,i}n_{i})^{2}}{k_{i}^{2}} ], (i = 1, 2,…, n). \end{aligned}  (2.15)

The equation (2.14) gives

[−ρ_{i} − (4k^{2} − 1)k_{i}u_{i}]M^{(2)}_{2k,i} + [ρ_{i} + (4k^{2} − 1)k_{i+1}u_{i}]M^{(2)}_{2k,i+1}

− (4k^{2} − 1)k_{i}N^{(2)}_{2k,i} + (4k^{2} − 1)k_{i+1}N^{(2)}_{2k,i+1}

= \frac{(k^{2} − 1)^{2}}{2k^{2}ρ_{i}} [\frac{(1 + k_{i+1}^{2})(k_{i+1}N^{(1)}_{k,i+1} − M^{(1)}_{k,i+1}n_{i+1})^{2}}{k_{i+1}}

− \frac{(1 + k_{i}^{2})(k_{i}N^{(1)}_{k,i} − M^{(1)}_{k,i}n_{i})^{2}}{k_{i}} ]

+ \frac{(4k^{2} − 1)(k^{2} − 1)^{2}}{2k^{2}ρ_{i}} [−(1 + k_{i+1}^{2})(k_{i+1}N^{(1)}_{k,i+1} − M^{(1)}_{k,i+1}n_{i+1})^{2}

+ (1 + k_{i}^{2})(k_{i}N^{(1)}_{k,i} − M^{(1)}_{k,i}n_{i})^{2}]

+ \frac{k_{i} − k_{i+1}}{ρ_{i}k^{2}} [(k^{2} − 1)^{2}(k_{i}^{2} − k^{2} − k_{i}k_{i+1}k^{2})(M^{(1)}_{k,i}u_{i} + N^{(1)}_{k,i})^{2}

+ (1 + k^{2})ρ^{2}M^{(1)}_{k,i}^{2} + ρ_{i}(k^{2} − 1)(2k_{i} + k^{2}k_{i} − k^{2}k_{i+1})(M^{(1)}_{k,i}u_{i} + N^{(1)}_{k,i})M^{(1)}_{k,i}].

(2.16)

Solving the system (1.14′) we have

\begin{matrix}φ^{(2)} & _{0}(u) = ∫ ( A^{(2)}_{0}(u) − ρ′(u)B^{(2)}′_{0}(u)) du + M^{(2)}_{0}, \\ ψ^{(2)} & _{0}(u) = B^{(2)}_{0}(u), \\ χ^{(2)} & _{0}(u) = ρ(u)[∫ \frac{C^{(2)}_{0}(u)}{ρ^{2}(u)} du + N^{(2)}_{0}]. \end{matrix}  (2.17)

where A^{(2)}_{0}(u), B^{(2)}_{0}(u), C^{(2)}_{0}(u) are given by (1.14). According to (2.9) we have

M^{(2)}_{0,i} − M^{(2)}_{0,i+1} = [ A^{(2)}_{0,i+1}(σ) − A^{(2)}_{0,i}(σ)]σ + (k_{i} − k_{i+1})C^{(2)}_{0,i}(σ),  (2.18)

(i = 1, 2,…, n). From (2.9) we also have

N^{(2)}_{0,i} = N^{(2)}_{0,i+1}, (i = 1, 2,…, n).  (2.19)

From above exposed, the following theorem holds:

Theorem. Neccesary and sufficient condition for the toroid rotational surface generated by polygonal meridian to have a field of infinitesimal bendings of the

second order as an extension of the field \overline{z}^{(1)}(u, v), is the systems of linear equations (2.15, 16) and (2.18, 19) to be compatible.

REFERENCES

[1] S. E. Kon-Fossen, Nekotorye voprosy differencial’noĭ geometrii v celom, Fizmatgiz, Moskva 1959

[2] I. Ivanova Karatopraklieva, Nezhestkost vtorogo poryadka nekotoryh sostavnyh poverhnostei vrascheniya, Serdika 3 (1977), 159–167

[3] N. G. Perlova, Sliding deformations of first, second and third order of surfaces of revolution with edges bounded by a parallel circle (in Russian), Comment. Math. Univ. Carolinae 12 (1971), 807–823

[4] Lj. S. Velimirović, Beskonechno malye izgibaniya toroobraznoi poverhnosti vrascheniya s mnogougol’nym meridianom, to appear in Izvestiya VUZ-ov—Matematika

[5] K. M. Belov, O beskonechno malyh izgibaniyah toroobraznoi poverhnosti vrashceniya, Sib. Mat. Zhurnal 9, 3 (1968) 490–494

[6] I. Ivanova-Karatopraklieva, I. Kh. Sabitov, Surface deformation. I, J. Math. Sci., New York 70, 2 (1994) 1685–1716

[7] I. Ivanova-Karatopraklieva, I. Kh. Sabitov, Bending of surfaces. II, J. Math. Sci., New York 74, 3 (1995) 997–1043

[8] V. A. Aleksandrov, Remarks on Sabitov’s conjecture that volume is stationary under an infinitesimal bending of a surface (in Russian), Sib. Mat. Zhurnal 30, 5 (1989) 16–24

(received 27.01.1997., in revised form 10.07.1997.)

Faculty of Civil Engineering, Beogradska 14, 18000 Niš, Yugoslavia e-mail: vljubica@ziux.grafak.ni.ac.yu