334 MR. L. F. RICHARDSON: APPROXIMATE ARITHMETICAL SOLUTION 



plane and unacted upon by surface forces except at the edges, the displacement w 



^4 ^4 ^4 



normal to the plate satisfies the equation = ^-^ + 2 ^ + -^r = V, 4 w, the edge 



9ar oar 3y* By* 



conditions being such as to make the displacement purely normal to the undeformed 

 plane and x and y being co-ordinates in this plane. So that if a sheet of thin steel 

 " ferrotype " is taken and fixed at the edges in such a way as to make a double layer 

 of values of w there equal to the integral of the boundary conditions of the dam, then 

 the displacement w elsewhere would also be equal to the integral x f the complete 

 system of equations for the dam. The principal stresses in the dam are then equal 

 to the principal curvatures of the plate. For a method of measuring them, see two 

 letters to 'Engineering,' October 25 and November 1, 1907. 



Whenever one has to solve the equation V/x = 0, the form assumed by a piece of 

 postcard bent in the fingers will be worth considering. As an accurate experimental 

 method this would have the advantage over those of PEARSON and of WILSON and 

 GORE, that displacements are applied instead of forces, and that the resulting 

 displacement to be measured may be larger than theirs were. 



4'1"13. To return to the Conditions at the Base of the Dam. Prof. PEARSON 

 pointed out to me that the effect of the stresses in the base of the dam, at a distance 

 where the base of the dam subtends but a small angle, will be the same as that of any 

 other statically equivalent system over the base, in particular to the force-at-a-point 

 found by compounding the pressure of the water acting through the centre of pressure 

 with the weight of the dam acting through its centre of gravity. We may therefore, 

 ia imagination, remove the dam, leaving a horizontal plane with this force acting at a 

 certain point, the pressure of the water in front and no pressure behind. Suppose we 

 had the distribution of ^ corresponding to this system of surface forces. Then the 

 corresponding double set of values of ^ at a considerable distance from the dam 

 will differ exceedingly little from the true values, and if we keep them fixed and write 

 in the upper surface values on the dam we may adjust the numbers inside by successive 

 approximation to satisfy V 4 ^ = 0, and the result will then be exceedingly close to the 

 true integral in the neighbourhood of the dam. This is what has been done. 



From the linearity of V 4 and of the stress equations, it follows that if the stresses 

 corresponding to a number of solutions of V 4 ^; = 0, when added, give the true stress, 

 then the stresses derived from the sum of all these solutions will also be correct. 

 With the aim of providing distributions of ^ in the bedrock which shall enable 

 engineers to solve " PEARSON'S Dam Problem" for any shape of dam boundary, I have 

 considered the actual distribution of ^ in the bedrock as made up of the three 

 following parts, which must be added in the proper proportions as described below : 



(i.) The term ^ = -^gpz 3 = |z 3 . This gives 3a%e = gpz, and so combines with 

 the stress due to the weight of the bedrock to make s x = 0. 



(5i.) In a weightless bedrock. The stress function \fi 2 , due to a point force in any 

 direction at the origin. 



