.}_ EQUILIBRIUM OF A SPHERICAL ENVELOPE. 



Next find the distribution of stress in the spherical surface due to each 

 of these pairs of forces, and combine them at every point by the rules for the 

 composition of stress*. The result will be the actual distribution of stress, and 

 if any unknown forces have been introduced in the process, they will disappear 

 from the result 



The calculation of the resultant stress from the component stresses, when 

 these are given in terms of u asymmetrical spherical co-ordinates of different 

 systems, would be very difficult; I shall therefore shew how to effect the same 

 object by a method derived from Mr Airy's valuable paper, "On the strains 

 in the interior of beamsf." 



If we place the point of inversion on the surface of the sphere, the in- 

 verse surface is a plane, and if p^, p,^ and p n represent the components of 

 in the plane referred to rectangular axes, we have for equilibrium 



< I3 >> 



These equations are equivalent to the following : 



where F is any function whatever of x and y. 



The form of the function F cannot, in the case of a plane, be determined 

 from the equations of equilibrium, as strains may exist independently of ex- 

 ternal forces. To solve the question we require to know not only the original 

 strains, but the law of elasticity of the plane sheet, whether it is uniform, or 

 variable from point to point, and in different directions at the same point. 

 When, however, we have found two solutions of F corresponding to different 

 cases, we can combine the results by simple addition, as the expressions (equa- 

 tion 15) are linear in form. 



In the case of two forces acting on a sphere, let A, B (fig. 29) be the 

 points corresponding to the points of application in the inverse plane ; AP = ? 



* Rankin's Applied Mechanics, p. 82. 



+ Philosophical Transactions, 1863, Part I. p. 49. 



