EQUILIBRIUM OF A SPHERICAL ENVELOPE. 89 



electricity in the sheet. Hence, if such systems of lines exist, there must 

 be some singular points, at which all the lines of flow meet, and at which 



dH , 



is infinite. 

 ao, 



7TT 



If -y~- is nowhere infinite, there can be no systems of lines at all, 

 and if -TOT is infinite at any point, there is an infinite stress at that point, 



which can only be maintained by the action of an external force applied at that 

 point. 



Hence a spherical surface, to which no external forces are applied, must 

 be free from stress, and it can easily be shewn from this that, when the external 

 forces are given, there can be only one system of stresses ha the surface. 



This is not true in the case of a plane surface. In a plane surface, 

 equation (3) disappears, and we have only two differential equations connecting 

 the stresses at any point, which are not sufficient to determine the distribution 

 of stress, unless we have some other conditions, such as the equations of elas- 

 ticity, by which the question may be rendered determinate. 



The simplest case of a spherical surface acted on by external forces is that 

 in which two equal and opposite forces P are applied at the extremities of a 

 diameter. There will evidently be a tension along the meridian lines, com- 

 bined with an equal and opposite pressure along the parallels of latitude, and 

 the magnitude p of either of these stresses will be 



p 



where a is the radius of the sphere, P the force at the poles, and 6 the 

 angular distance of a point of the surface from the pole. If < is the longitude, 

 and if i\ and r, are the rectilineal distances of a point from the two poles 

 respectively, and if we make 



= log.^ and H=<f> ........................... (8), 



then G and H will give the lines of principal stress, and 



<D= - T?7 I = I-TT7 



27ra \ao 2 / 2va \ao,y 



VOL. II. 12 



