on Revetment Walls. 497 



friction exerted at any point will be exerted in some direction in 

 a vertical plane parallel to the revetment wall, will be seen from 

 what follows a little further on, toimply that every such plane 

 contains the radius vector which makes the greatest angle with 

 the normal, and consequently the section of the ellipsoid of stress 

 with which we are dealing will be the plane of greatest and least 

 thrust, or greatest and least pressure. By way of aid to the 

 imagination in seizing this subtle conception of stress (a real 

 conquest in physical ideology due to the last quarter of the 

 present century, although its first germ may be recognized in 

 the much earlier molecular view of the circumambient pressures 

 round about each internal point of a perfect fluid), 1 have gone 

 thus briefly into the generation of the ellipse and ellipsoid above 

 described ; but I shall have very little occasion, except for 

 occasional facility of reference, to have resort to them, as the 

 equations (1) and (2) will suffice for my purpose in the present 

 inquiry. 



These are the equations which govern the distribution of 

 stress; and it may be convenient to confer upon the ellipse 

 whose radii vectores are in length inversely as the square roots of 

 the pressures acting upon them, the name of the ellipse oi pres- 

 sures s in order to obviate any possibility of the position of this 

 ellipse being confounded with that of the one which would, I 

 believe, more ordinarily go by the name of the ellipse of stress. 

 Every point in the mass is the centre of such an ellipse ; and 

 those ellipses, if properly drawn, will represent completely, and 

 on the same scale, the magnitude and distribution of the pres- 

 sures round about any point. It is almost needless to add that 

 for a perfect fluid these ellipses would become circles. 



Let us now proceed to establish the law of the variation of 

 the stresses, or, to speak more accurately, of the thrusts acting 

 on planes drawn in any given directions, on passing from one 

 point of the mass to another. Returning to our little rect- 

 angular element PQRS, and considering the lines PQ, PS to be 

 given in direction, so that we may consider PQ = cfo? and PS — dy, 

 and calling the units of pressure on PQ and RQ L and N, 

 the unit of face-force M, the impressed forces of acceleration X 

 in the direction of x, and Y in the direction of y, and the unit of 

 mass p, by simple estimation of the forces in the directions of x 

 and y respectively we obviously obtain, due attention being paid 

 to the mode of fixing the positive directions of X and Y, 



dh dM_ 



dy dx ~~ * ' 



Jn dM_ x 



dx dy ~~ 



