TRANSFORMATION OF SURFACES BY BENDING. 



453 



On the properties of a Surface considered as the limit of the inscribed Polyhedron. 



1. To inscribe a polyhedron in a given surface, all whose 

 sides shall be triangles, and all whose solid angles shall be 

 hexahedral. 



On the given surface describe a series of curves according 

 to any assumed law. Describe a second series intersecting these 

 in any manner, so as to divide the whole surface into quadri- 

 laterals. Lastly, describe a third series (the dotted lines in the 

 figure), so as to pass through all the intersections of the first 

 and second series, forming the diagonals of the quadrilaterals. 



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 : — 



- If 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 immove- 

 able 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 directions. 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. 



Aug. 30, 1854. 



J. C. M. 



