on Revetment Walls. 499 



L=P(cos0) 9 + R(sin0) 2 , 

 M=(P-R)sin0cos0, 



M 



tan (\) == (fi) = maximum value of -j-. 



To find this maximum, we have 



8{cot0 + 7 2 tan0}=O. 

 Hence 



7 tan 0=1, 

 and therefore 



2 7 



cot (X) = 



1-7 2 ' 

 therefore 



(l-7 2 )-2 7 tan\=0, 

 or 



. 7 = 8 ec(X)-tan(X) = I^Uan( 4 5-^). 



This equation expresses the universal relation between the 

 form of the ellipse of pressures for any stress and the relative 

 angle of repose for such stress. 



The problem we have just solved may be presented advan- 

 tageously, in order to make the impression of it more vivid (as 

 it is of cardinal importance), under a geometrical point of view. 

 Taking any radius vector of the ellipse of pressures, the angle 

 between it and its conjugate radius is 90° at any vertex ; at some 

 point therefore it will be at a minimum, and this minimum 

 will be the complement of the relative angle of repose. 



From the preceding investigation, it will easily be seen that, 

 to find the ray-directions which give this minimum, we have 

 only to construct a rectangle circumscribing the ellipse, and 

 either of its two diagonals will be in the direction required, and 

 the angle between either such ray and the principal axes plus 

 or minus half the angle between it and the normal (which angle 

 is the relative angle of repose) will be half a right angle*. 



* In fact the diameters which coincide with the directions of these 

 diagonals are conjugate diameters, equally inclined to the principal axes ; 

 and these, as I suppose must be well known, are the conjugate diameters 

 whose inclination to each other is a minimum. 



[To be continued.] 



