TRANSFORMATION OF SURFACES BY BENDING. 91 



NOTE. Since this paper was written, I have seen the Rev. Professor Jellett's Memoir, On 

 the Properties of Inextensible Surfaces. It is to be found in the Transactions of the Royal Irish 

 Academy, Vol. XXII. Science, &c., and was read May 23, 1853. 



Professor Jellett has obtained a system of three partial differential equations which express 

 the conditions to which the displacements of a continuous inextensible membrane are subject. 

 From these he has deduced the two theorems of Gauss, relating to the invariability of the product 

 of the radii of curvature at any point, and of the " entire curvature" of a finite portion of the 

 surface. 



He has then applied his method to the consideration of cases in which the flexibility of the 

 surface is limited by certain conditions, and he has obtained the following results : 



If the displacements of an inextensible surface be all parallel to the same plane, the surface 

 moves as a rigid body, 



Or, more generally, 



If the movement of an inextensible surface, parallel to any one line, be,' that of a rigid body, the 

 entire movement is that of a rigid body. 



The following theorems relate to the case in which a curve traced on the surface is rendered 

 rigid : 



// any curve be traced upon an inextensible surface whose principal radii of curvature are finite 

 and of the same sign, and if this curve be rendered immoveable, the entire surface will become 

 immoveable also. 



In a developable surface composed of an inextensible membrane, any one of its rectilinear 

 sections may be fixed without destroying the flexibility of the membrane. 



In convexo-concave surfaces, there are two directions passing through every point of the 

 surface, such that the curvature of a normal section taken in these directions vanishes. We 

 may therefore conceive the entire surface to be crossed by two series of curves, such that 

 a tangent drawn to either of them at any point shall coincide with one of these direc- 

 tions. These curves Professor Jellett has denominated Curves of Flexure, from the following 

 properties : 



Any curve of flexure may be fixed without destroying the flexibility of the surface. 



If an arc of a curve traced upon an inextensible surface be rendered fixed or rigid, the entire of 

 the quadrilateral, formed by drawing the two curves of flexure through each extremity of the curve, 

 becomes fixed or rigid also. 



Professor Jellett has also investigated the properties of partially inextensible surfaces, and 

 of thin material laminae whose extensibility is small, and in a note he has demonstrated the 

 following theorem : 



If a closed oval surface be perfectly inextensible, it is also perfectly rigid. 



A demonstration of one of Professor Jellett's theorems will be found at the end of this paper. 



J. C. M. 

 Aug. 30, 1854. 



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