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



sible (as hereafter explained), and how far the preceding theorems are appli- 

 cable to such surfaces ; and, finally, I shall consider how far these conclusions 

 are applicable to the laminas which we find in nature, which are neither wholly 

 inextensible nor wholly devoid of thickness. The results arrived at will be 

 found, I think, sufficiently remarkable to attract the attention of mathematicians 

 to this subject. 



2. Definition of an Inextensible Surface. — Two definitions of inextensibility 

 have been given by Lagrange and Gauss respectively. According to the 

 former, who defines the force which resists extension to be the force which resists 

 the increase of superficial area, a surface is inextensible if it be impossible to 

 change the superficial area of any portion of it. But this definition seems to 

 be hardly consistent with the meaning ordinarily attached to the word " inex- 

 tensible." For if we conceive a membrane admitting of being indefinitely ex- 

 tended in any direction, but of such a nature, that an extension in any one 

 direction is always accompanied by a corresponding contraction in another, so 

 as to preserve the area unchanged, such a membrane wovdd be, according 

 to Lagrange's definition, inextensible. But it appears more consistent with 

 ordinary ideas to consider an inextensible surface to be one which does not 

 admit of any extension, rather than one whose capacities of extension and con- 

 traction counterbalance one another in the manner above described. I shall, 

 therefore, in the present Memoir adopt the definition of Gauss, as more ex- 

 actly embodying the ordinary ideas on the subject, adding to it the definition 

 of partially extensible surfaces, a class not noticed by Gauss, but jiresenting 

 some remarkable properties. These definitions are as follows : 



I. A suifuce is said to be inextensible, when the length (fa curve traced arbi- 

 trarily ujKin it is unchangeable by any force which can be applied to it. 



II. A surface is partially extensible, if there be at each of its points one or 

 more inextensible directions; in other words, if it be possible to trace at each point 

 one or more inextensible curves. 



We shall now proceed to consider how these definitions' may be mathe- 

 matically expressed, commencing with the case of inextensible sui-faces. 



3. Deduction of the Equations which connect the Displacements of an Inex- 

 tensible Surface. Let ds be tlie element of a curve traced in any direction upon 

 the surface, and let c be the symbol of displacement, i. e. a symbol denoting the 



