The Rev. J. H. Jellett on the Properties of Inextensible Surfaces. 367 



8. The preceding discussion of the properties of inextensible surfaces is 

 of course a mathematical abstraction, not strictly applicable to any substance 

 which we find in nature. Every membrane with which we are acquainted is 

 possessed of some extensibility ; and all substances have of course a certain 

 thickness. Our definition, therefore, of an inextensible surface is not strictly 

 true for any really existing substance. But as there are in nature many sub- 

 stances for which this definition is very approximately true, it becomes a question 

 of some interest to determine how far the results of the preceding investi- 

 gation are applicable to such substances. We shall, therefore, proceed to con- 

 sider the case of a membrane whose thickness is indefinitely small as compared 

 with its other dimensions, and whose extensibility is such that in any dis- 

 placement of the membrane, the variation in the length of any arc of a curve 

 traced upon its surface is indefinitely small compared with the displacement 

 of any of its parts. Thus, if x, y, z be the co-ordinates of any point on the sur- 

 face, and s an arc of a curve traced upon it, the assumption which we shall 

 make as to the inextensibility of the membrane may be mathematically ex- 

 pressed by saying that Es is indefinitely small compared with ex. If it be ne- 

 cessary to take thickness into account, we must suppose s to be a curve traced 

 arbitrarily in the substance of the membrane. Supposing, for the sake of greater 

 generality, that this is the case, we may state the problem under discussion 

 as follows : 



To determine the possible displacement of a membrane very slightly exten- 

 sible, and whose thickness is very small compared with its other dimensions. 



Let x', y', z' be the co-ordinates of a point in the substance of the mem- 

 brane ; X, y, z, the co-ordinates of a point on the surface, indefinitely near to 

 the first ; and i, a quantity of the same order of magnitude as the thickness of 

 the membrane. 



Through the point x'y'z' let a normal be drawn to the surface of the mem- 

 brane, and let n represent the part of the normal between x'y'z' and its inter- 

 section with the surface, which we shall denote by x, y, z. Then if a, p, y be 

 the cosines of the angles which the normal makes with the axes, we shall have 



x' = X + a7l, 



z' = z + yn. 



