364 The Eev. J. H. Jellett on the Properties of In extensible Surfaces. 



Let TT'be an arbitrary function of a- and y. Then the equations (A') being 

 put under tlie form 



^^TT>, ^Ji + ^ = 2Ws, J^=m, 



d.r ' dy dx dy 



the expression for the extension of any small arc ds (Y) will become 



ids = {W- w) ds (r cos^ a + 2s cos a cos^ + t cos-/3). 



Hence for the class of surfaces under consideration we infer that — 



The extension of any small arc of a airve commencing at a given point, divided 

 by the arc itself varies inversely as the radim of curvature of the normal section 

 which passes through it. 



7. Having thus investigated the case of inextensible and partially inexten- 

 sible surfaces, we should, in the next place, proceed to consider how far the 

 results arrived at are applicable to the various membranes which we find in 

 nature, and which are neither perfectly inextensible nor altogether devoid of 

 thickness. But before entering upon this question we shall briefly examine 

 the case of inextensible bodies. 



Conceive a curve to be traced in the interior of a bod}', passing through the 

 successive physical points or molecules a, b, c, d, &c. Suppose now that the se- 

 veral points of the body receive small displacements, and take the curve which 

 is the locus of the points a, b, c, d, &c. in their new position. If the length of the 

 second curve be equal to that of the first, and if this be true of all curves which 

 can be so drawn, the body may be said to be inextensible. Adopting this defi- 

 nition, we shall liave the following theorems : — 



I. Every body which is perfectly inextensible is also perfectly rigid. 



II. Any body may, without being wholly inextensible, have at each of its points 

 an infinite number- of inextensible directions, and these directions icill be situated upon 

 a cone of the second order. 



Let Ix, ly, iz, be the displacements of any point in the body, and let ds be 

 an element of a curve, making with the axes of co-ordinates the angles a, fi, y. 

 Then it is easily seen, that the variation of this element will be given by the 

 equation 



