<JU EQUILIBRIUM OF A SPHERICAL ENVELOPE. 



To pass from this case to that in which the two forces are applied at 

 any points, I shall make use of the following property of inverse surfaces. 



If a surface of any form is in equilibrium under any system of stresses, 

 and if lines of principal stress be drawn on it, then if a second surface be the 

 i morse of the first with respect to a given point, and if lines be drawn on 

 it which are inverse to the lines of principal stress in the first surface, and 

 if along these lines stresses are applied which are to those in the corresponding 

 point of the first surface inversely as the squares of their respective distances 

 from the point of inversion, then every part of the second surface will either 

 be in equilibrium, or will be acted on by a resultant force in the direction 

 of the point of inversion. 



For, if we compare corresponding elements of lines of stress in the two 

 .surfaces, we shall find that the forces acting on them are in the same plane 

 with the line through the point of inversion, and make equal angles with it. 

 The moment of the force on either element about the point of inversion is 

 therefore as the product of the length of the element, into its distance 

 from the point of inversion into the intensity of the stress. But the length 

 of the element is as its distance, and the stress is inversely as the square of 

 the distance, therefore the moments of the stresses about the point of inversion 

 are equal, and in the same plane. If now any portion of the first surface is 

 in equilibrium, it will be in equilibrium as regards moments about the point 

 of inversion. The corresponding portion of. the second surface will also be in 

 equilibrium as regards moments about the point of inversion. It is therefore 

 either in equilibrium, or the resultant force acting on it passes through the 

 point of inversion. 



Now let the first surface be a sphere ; we know that the second surface 

 is also a sphere. In the first surface the condition of equilibrium normal to 

 the surface is p a +p a = 0- In the second surface the stresses are to those 

 in the first in the inverse ratio of the squares of the distances. Hence in 

 the second surface also, p n + p a = 0, or there is equilibrium in the direction of 

 the normal. But we have seen that the resultant, if any, is in the radius 

 vector. Therefore, if we except the limiting case in which the radius vector 

 is perpendicular to the normal, the equilibrium is complete in all directions. 



We may now, by inverting the spherical surface, pass from the case of 

 a sphere acted on by a pair of tensions applied at the extremities of a 



