gg EQUILIBRIUM OF A SPHERICAL ENVELOPE. 



would have been the tensions in the principal planes of curvature, and not 

 necessarily principal tensions. 



In the case of a spherical surface not acted on by any fluid pressure, 

 r t *T v and JV, = A" t = 0, so that the third equation becomes 



Pu+P* = (-4), 



whence we obtain from the first and second equations 



where <7, is a function of H, and C t of G. If then we draw two lines of 

 the system (//= constant) at such a distance that p u (dS^ = (dh)* at any point 

 where (dh)' is constant, this equation will continue true through the whole 

 length of these lines, that is, the principal stresses will be inversely as the 

 square of the distance between the consecutive lines of stress. Since this is 

 true of both sets of lines, we may assume the form of the functions G and 

 H, so that they not only indicate lines of stress, but give the value of the 

 stress at any point by the equations 



/dH* dG\* 



where (//= constant) is a line of principal tension, and (G = constant) a line of 

 principal pressure. 



If we now draw on the spherical surface lines corresponding to values of G 

 differing ' by unity, and also lines corresponding to values of H differing by 

 unity, these two systems of lines will intersect everywhere at right angles, and 

 the distance between two consecutive lines of one system will be equal to the 

 distance between two consecutive lines of the other, and the principal stresses 

 will be in the directions of the lines, and inversely as the square of the 

 intervals between them. 



Now if two systems of lines can be drawn on a surface so as to fulfil 

 these conditions, we know from the theory of electrical conduction in a sheet 

 of uniform conductivity, that if one set of the curves are taken as equipotential 

 lines, the other set will be lines of flow, and that the two systems of lines 

 will give a solution of some problem relating to the flow of electricity through 

 a conducting sheet. But we know that unless electricity be brought to some 

 point of the sheet, and carried off at anoUier point, there can be no flow of 



