BY FINITE DIFFERENCES OF PHYSICAL PROBLEMS, ETC. 347 



into the boundary equations, in that the average of (11) and (10) is derived from 

 the average of (10) and (11) by means of the value of dx/dq at the centre of the four 



points where these values are situated. 



(c) (11) is derived from (11) by means of the value of dx/dq half-way between 

 them and (21) is similarly derived from (12). But (01) is derived from (10) by c^xi^l 

 at \, J and (10) is also derived from (10) by 3^/3</ at 00, where in the latter q is 



normal to the line joining 01 and 00. Thus (10) and (Ol) are both connected to (10), 

 and are no longer independent. It is therefore impossible to satisfy the body equations 



completely. Large values of ^i*x may appear at 10 and 01. In some work by this 



method, now rejected, the approximation was so arranged 



that these values of \? i\ tended to become equal and TABLE XIV 



opposite. The reason for choosing (a) was that the system (c) 



gave large oscillations in the stresses near the corners, 



oscillations for which it was difficult to find any physical 



reason ; (b) gave extraordinary oscillations in the shears on 



the boundary, but (a) gave smooth curves of stress. _ 



To confirm this choice the co-ordinate differences have been 

 altered so that the boundary lies half-way between two 

 tabular numbers. The ambiguities now disappear. Thus, at 



the hind toe it is found on stepping out the integral of boundary equations that we 

 have zero values, as in Tables X. and XIV., whatever be the size of the co-ordinate 

 difference. 



TV X a v a ,, 



Then at the corner -A = - _JL = - _ . Therefore - 



hx* 2 So? %z 2 



Now test Table XII. to see whether it gives the same result. 



At the hind toe |* = + 3'88, nM.f$r = -379, and - = 1-02, which is 

 So: 'ftxftz 379 



satisfactory. 



At the front toe a more elaborate check has been made by means of a new table 

 with smaller co-ordinate differences, h = %. This is Table XV. It was prepared as 

 tollows : First, the number of digits in the values of x, near the corner, were reduced 

 by taking instead - x ' = -x-74 + 30a;-18z, so that x and x' have identical second- 

 differences. Next, curves were plotted showing the variation of x' along lines parallel 

 to the co-ordinate axes. Values of x' were read from the curves at the new points 

 required. These values provided the boundary numbers of Table XV., on the side of 

 it which is in the middle of the dam. They also provided the initial body values, 

 which are given in parentheses in Table XV. The boundary numbers on the 

 masonry-water surface were, however, not obtained by interpolation from Table XII., 

 but instead from the infinitesimal integral of the boundary conditions. Each of them 



2 Y 2 



