[Extracted from The Quarterly Journal of Pure and Applied Mathematics, No. 32, 1867.] 



XXX. On the Equilibrium of a Spherical Envelope. 



I PROPOSE to determine the distribution of stress in an indefinitely thin 

 und inextensible spherical sheet, arising from the action of external forces applied 

 to it at any number of points on its surface. 



Notation. Let two systems of lines, cutting each other at right angles, be 

 drawn upon any surface, and let their equations be 



G and <> t xz = H, 



where each curve is found by putting G or H equal to a constant, and com- 

 bining it with the equation of the surface itself, which we may denote by 



Now let G be made constant, and let H vary, and let dS 1 be the element 

 of length of the curve (G = constant) intercepted between the two curves for 



which // varies by dH, then -TTT will be a function of H and G. 



ao, 



In the same way, making dS t an element of the curve (//= constant) we 

 may determine -j-^ as a function of H and G. 



Now let the element dS t experience a stress, consisting of a force X in 

 the direction in which G increases, and Y in the direction in which H in- 

 creases, acting on the positive side of the linear element dS u and equal and 

 opposite forces acting on the negative side. These will constitute a longitudinal 

 tension normal to dS lt which we shall denote by 



X 



