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each point of the surface two directions, such that the normal 

 sections which pass through them respectively have at that 

 point infinite radii of curvature. We may, therefore, con- 

 ceive the entire surface to be crossed by two series of curves, 

 such that if at any point a tangent be drawn to either of the 

 curves which pass through it, the normal section passing 

 through this tangent will have at that point an infinite radius 

 of curvature. These curves we shall denominate (for a reason 

 which will appear subsequently) curves of flexure. Preserv- 

 ing this definition, we shall have the following theorems : 



I. If an arc of a curve (which is not a curve of flexure), 

 traced upon an inextensible surface of the third class, be rendered 

 immovable, and if we conceive the two curves of flexure corres- 

 ponding to the extreme points of the fixed arc to be drawn, the 

 whole of the quadrilateral formed by these four curves will become 

 immovable also. (In forming this quadi'ilateral it is to be re- 

 membered that each of its angular points is formed by the in- 

 tersection of two curves belonging to different series.) 



II. A curve of flexure may be fixed, without rendering any 

 finite portion of the surface immovable. 



III. If two arcs of curves of flexure, commencing from the 

 same point, be fixed, the immovable portion of the surface will be 

 the quadrilateral formed by these two arcs, and the two other 

 curves of flexure drawn through their other extremities. 



The preceding theorem (II.) gives the reason for the name 

 " curves of flexure." In fact, we see that if one of these curves 

 be fixed, the surface has the power of bending round it. This 

 would be impossible with any other curve. 



The author has next proceeded to consider the case of sur- 

 faces which may be denominated partially extensible. These 

 surfaces have at each of their points one or more inextensible 

 directions. In other words, it is possible to draw through 

 each point of the surface one or more inextensible curves. 

 Respecting these surfaces, the author has arrived at the follow- 

 ing results : 



