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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. [fan are 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 are to be drawn, the 
whole of the quadrilateral formed by these four curves will become 
immovable also. (In forming this quadrilateral 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 flecure may be fixed, without rendering any 
finite portion of the surface immovable. 
III. Jf two arcs of curves of flexure, commencing from the 
same point, be fied, 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 : 
