490 Prof. Sylvester on the Pressure of Earth 



evidently the following : — " Of all the possible states of equili- 

 brium of the earth consistent with the assigned conditions, to 

 determine that one which shall make the greater of two quan- 

 tities to be named the least possible," — one of these quantities 

 being the thickness of the wall determined by the condition 

 that its friction with the ground shall be just equal to the sum 

 of the horizontal pressures on the wall, the other by the condi- 

 tion that its moment about the edge most remote from the 

 earth shall be just equal to the sum of the moments of the 

 entire thrust upon the wall at each several element thereof in 

 respect to the same edge. 



Whenever Coulomb's method leads to a right solution of the 

 problem of revetments, the thrusts on the several elements of the 

 wall will be all parallel ; and it may easily be seen that, in solving 

 the problem for this case, we are solving the problem of making 

 the statical sum of the thrusts a minimum ; and the result will 

 be the same, whether the pier can only be pushed bodily on its 

 base, or can only turn over an edge, or can do both one and the 

 other. But it must obviously be erroneous to assume as a uni- 

 versal principle, that in the state bordering upon motion, or what 

 is going still further, in a state antecedent to this, the statical 

 sum of the pressures will be a minimum ; and if Mr. Moseley's 

 " principle of least resistance," quoted by Professor Rankine, 

 means this, I have no scruple in proclaiming my entire dissent 

 from such an assumption. I do not here enter at all into the 

 question of determining pressure, except in the state of equili- 

 brium bordering upon motion ; and in that state common sense 

 points out that it is not the pressure or sum of pressures, but the 

 effect of such pressure or pressures in inducing motion in a certain 

 possible manner, or in any one out of a choice of possible 

 manners, that governs the determination of the minimum. This 

 principle of least resistance is one of the shoals upon which Mr. 

 Rankine's investigation appears to me to have split. 



Be it observed that the only physical assumption which I pro- 

 pose is this, that if equilibrium can be preserved consistently with 

 the imposed conditions, equilibrium will be preserved. Without 

 such a supposition the question would be incapable of treatment 

 without further laws regulating the interior forces than we 

 suppose given. The legitimacy of such an assumption cannot, I 

 think, be seriously called into question, and once made, the 

 problem of determining the wall's thickness becomes a purely 

 mathematical question ; one undoubtedly of great difficulty, 

 but perfectly determinate, and falling under the dominion of 

 the Calculus of Variations, as will easily be recognized from the 

 circumstance that the integration of the general equations of 

 equilibrium, if it could be performed, would necessarily contain 



