THE 



LONDON, EDINBURGH and DUBLIN 



PHILOSOPHICAL MAGAZINE 



AND 



JOURNAL OF SCIENCE. 



SUPPLEMENT to VOL. XX. FOURTH SERIES. 



LXVL On the Pressure of Earth on Revetment Walls* By 

 J. J. Sylvester, Professor of Mathematics at the Royal 

 Military Academy*. 



Part I. Critique of the Hypothesis of Parallel " Planes of 

 Rupture." 



THE ensuing investigation deals with the pressure of Mathe- 

 matical earth. By mathematical earth, I mean earth 

 treated according to the idea of Coulomb, viz. as a continuous t 

 mass separable by planes in all directions, but whose separating 

 surfaces exert upon one another forces consisting of two parts, 

 one of the nature of ordinary friction, the other of so-called 

 cohesion. Of the latter, for greater simplicity, I shall com- 

 mence with taking no account, so that the matter with which 

 we have to deal becomes, so to say, " a frictional fluid." If we 

 isolate in idea any element of this fluid — suppose, to fix the 

 ideas, a molecule bounded by plane faces, this molecule will be 

 kept at rest by its own weight, the pressures on the several 

 faces, and the forces of friction acting along these faces : these 

 last-named forces are limited not to exceed the product of the 

 corresponding pressures by a certain coefficient, termed the co- 

 efficient of friction. 



In order to render the inquiry before us quite definite, let us 

 begin with supposing two vertical side walls and a back of solid 

 immoveable masonry, between which the earth is piled up in a 

 determinate form, fronted by a pier of given specific gravity, 

 whose minimum thickness is to be determined by the condition 

 that it may just suffice to prevent the pier from being either 

 forced forward or turned round over its further edge. The 

 earth is thus of course supposed to have only one free face, being 

 entirely supported at the sides and the back by the masonry 

 just spoken of. The problem then that we have to solve is 



* Communicated by the Author. 



t The only essential quality of our mathematical earth which differen- 

 tiates it from actual vulgar earth is this of continuity. 



Phil. Mag. S. 4. No. 136. Suppl. Vol. 20. 2 K 



