186 



The fundamental principle of the internal stability of such a mass 

 has already been published in the ' Proceedings of the Royal Society' 

 for the 6th of March, 1856, viz. that the ratio of the difference to 

 the sum of the greatest and least pressures at each point of the mass 

 must not exceed the sine of the angle of repose. 



The principles of the general theory of the internal equilibrium of 

 a solid mass are expressed in a form suited to the special subject of 

 the paper. For the purpose of determining the conditions of equi- 

 librium under its oven weight, of a solid mass whose upper surface is 

 that generated by the motion of a horizontal straight line along a 

 line of any figure described on a vertical plane at right angles to the 

 generating line, the mass is supposed to be divided into layers of 

 equal horizontal thrust by a series of surfaces, which layers are sub- 

 divided into elementary horizontal prisms by vertical planes normal 

 to the vertical plane first mentioned. For independent variables 

 there are taken the horizontal coordinate in this plane, and the total 

 horizontal thrust from the upper surface down to a given surface of 

 equal thrust. The condition of equilibrium of any one of the be- 

 fore-mentioned elementary prisms being expressed by a differential 

 equation in terms of those variables, the integration of that equation 

 gives the vertical coordinate of any surface of equal thrust in terms 

 of the total thrust down to that surface and of the horizontal co- 

 ordinate. The integral obtained belongs to a class first investigated 

 by Fourier. 



An approximation to the forms of the surfaces of equal thrust 

 is obtained by a simple graphic process, first employed by Prof. 

 "William Thomson in connexion with the theory of electricity. 



It is shown incidentally how the same integral may be applied to 

 determine the intrados from the extrados of any arched rib, loaded 

 only with its own weight. 



The pressure on a surface of equal thrust is vertical ; the pressure 

 on a vertical plane at a given point is parallel to the surface of equal 

 thrust traversing that point. When the upper surface of the mass 

 of earth is one plane, horizontal or inclined, the surfaces of equal 

 thrust are planes parallel to it. When the upper surface presents 

 elevations and depressions, the surfaces of equal thrust have corre- 

 sponding elevations and depressions, gradually vanishing as the 

 depth increases. 



