343 



XV. — On the Properties of InextensiUe Surfaces. By the Rev. John II. Jellett 

 A. M., Fellow of Trinity College, and Professor of Natural Philosophy 

 in the University of Dublin. 



Read May 23, 1853. 



1. Although the celebrated theorems ofGAUSS have received from mathe- 

 maticians much and deserved attention, inducing them to bestow considerable 

 labour upon obtaining for these theorems simple and elegant demonstrations 

 I do not find that any attempt has been made to extend his discoveries upon 

 this subject. Yet the highly interesting character of the theorems alluded to 

 might naturally induce the expectation of other important results connected with 

 the theory of inextensible surfaces, sufficient to repay the labour of a more 

 general consideration of the question than has been (so far as I am aware) as 

 yet attempted. I propose, therefore, in the present Memoir to consider gene- 

 rally what are the conditions to which the displacements of a continuous inex- 

 tensible membrane are subject. These conditions are expressed (as will be 

 seen) by a system of three partial differential equations of a very simple form, 

 which contain the solution of all questions connected with this theory. From 

 these equations I shall deduce general expressions for the variations which 

 the differential coefficients, 



dz dz d'z d-z d'^z 

 di' d^' d?' d^y' If' 



undei'go in consequence of the displacement of the membrane. These expres- 

 sions give immediately the two theorems of Gauss. I shall then proceed to 

 consider how far the flexibility of the membrane is destroyed by rendering 

 rigid any curve traced upon its surface. I shall in the next place investigate 

 the laws which govern the displacement of a surface which is partially exten- 



