EQUILIBRIUM OF A SPHERICAL ENVELOPE. 87 



and a shearing force on the element, which we shall call 



T 



In like manner, if the element dS 3 is acted on by forces Y' and X', it 

 will experience on its positive side a tension and a shearing force, the values 

 of which will be 



=J r and = x*_ 



That the moments of these forces on the elements of area dS 1} dS t may 

 vanish, 



or the shearing force on dS^ must be equal to that on dS^. 



When there is no shearing force, then p n and p^ are called the principal 

 stresses at the point, and if p a vanishes everywhere, the curves, (G = constant) 

 and (H= constant), are called lines of principal stress. In this case the con- 

 ditions of equilibrium of the element dS^dS^ are 



, , } - = () (2) 



dHdG +(P " p "> dGdH 



Py+P^^N.-N, (3). 



r t r t 



The first and second of these equations are the conditions of equilibrium 

 in the directions of the first and second lines of principal tension respectively. 



The third equation is the condition of equilibrium normal to the surface ; 

 r, and r t are the radii of curvature of normal sections touching the first and 

 second lines of principal stress. They are not necessarily the principal radii of 

 curvature. JV, is the normal pressure of any fluid on the surface from the side 

 on which r, and r a are reckoned positive, and N., is the normal pressure on the 

 other side. 



If the systems of curves G and H, instead of being lines of principal stress, 

 had been lines of curvature, we should still have had the same equation (3), 

 but r l and r, would have been the principal radii of curvature, and p n and p^ 



