94 EQUILIBRIUM OF A SPHERICAL ENVELOPE. 



Since ^' + ^0 in this case, the arbitrary functions must be zero, and 



djf dtf 

 we have to find the value of F from that of G by ordinary integration of (21). 



The result is 



..r if the co-ordinates of A and B are (a u 6,) and (o 6,), 

 Pain a f. (2x-a l -a,)(a,-a 1 ) + (2y-6.-6,)(6.-6 l ) . 

 - - ' - g 







x-Oi/J 



If we obtain the values of F for all the different pairs of forces acting 

 on the sphere, and add them together, we shall find a new value of F, the 

 second differential coefficients of which, with respect to x and y will give a 

 system of components of stress in the plane, which, being transferred to the 

 sphere by the process of inversion, will give the complete solution of the 

 problem in the case of the sphere. 



We have now solved the problem in the case of any number of forces 

 applied to points of the spherical surface, and all other cases may be reduced 

 to this, but it is worth while to notice certain special cases. 



If two equal and opposite twists be applied at any two points of the 

 sphere, >we can determine the distribution of stress. For if we put 



and JT=log- 1 -^ ..................... (26), 



the equations of equilibrium will still hold, and the principal stresses at any 

 point will be inclined 45 to those in the case already considered. 



If M be the moment of the couple in a plane perpendicular to the chord, 

 the absolute value of the principal stresses at any point is 



In figure 30 are represented the stereographic projections of the 

 lines of stress in the cases which we have considered. When a tension is 

 applied along the chord AB, the lines of tension are the circles through AB, 



