on Revetment Walls. 495 



N=R(cos<9) 2 + P( 8 in0) 2 , (1) 



F=s(R-P)sin0.'cos0; (2) 



showing that in elements in the directions of the principal axes 

 the face-forces vanish, and the thrusts become purely pressures, 

 I. e. forces perpendicular to the surfaces upon which they act. 

 R and P are of course essentially positive, as otherwise the mo- 

 lecules would be subject to a force of separation instead of 

 compression, and consequently the conic in question is an ellipse. 

 The total value of the thrust = \/N 2 +F 2 



=V / R 2 (cos(9) 2 + P 2 ( sm ^) 2 - ....(*) 

 R and P will evidently be in the directions in which, for a 

 given point, the entire thrust, as well as the pressure-part of it, 

 are the least and greatest. These directions may be said to be 

 those of " principal thrust." If we start from any point and 

 proceed from that point always in the direction of a line of 

 principal thrust so as to form a continuous curve, two such curves 

 cutting each other at right angles will intersect every point of 

 the mass at rest, of which, in the case of mathematical earth, I 

 may state, by way of anticipation, that only one can cut the free 

 surface when that surface is supposed to form part of a horizontal 

 plane. 



These lines may also be termed the principal lines of pressure, 

 or simply the lines of pressure ; and this name may be considered 

 indifferently to have reference either to the fact that the thrust 

 in the direction of the tangent at any point in any such curve is 

 the thrust acting upon the normal, or to the fact that the thrust 

 upon the tangent at any point is in the direction of the normal; 

 as either one of such conditions implies the other. 



The cosine of the angle between the pressure and the thrust 

 will be 



R(cosfl) 2 + P(sinfl) 2 



\ZWJcwdf+¥* (sin Of ' 



which, calling the principal semiaxes of the ellipse referred to a 

 and b respectively, and the rectangular coordinates of any point 

 therein x and y, becomes 



x* y 1 



a^ + b* 1 



which is equal to the perpendicular from the centre on the tan- 

 gent divided by the radius vector, showing that the direction of 

 the thrust on any radius of the ellipse in question is in the 

 direction of the conjugate diameter, whereby it is seen that the 



