342 ME. L. F. EICHAEDSON : APPEOXIMATE AEITHMETICAL SOLUTION 



greater than for the gravest mode which has X^ = 0'15 ; it will probably be as great 

 as unity. Then the errors in x must be roughly equal to ^ 4 ^. If we imagine the 

 tabulated values of ^ 4 x smoothed to eliminate all vibrations, except the one with the 

 single nodal line already referred to, and if we then take second differences of the 

 smoothed ^ 4 x, these second differences will be the errors in the stresses. It is easy 

 to see that they will be less than unity. 



S 4 '2 "3. Halving the Differences. Some notion of the error due to having so few as 

 4 co-ordinate differences in the base of the dam may be formed from the error in & 4 , 

 found from the equation (^ 4 k*) <f> = 0, for the gravest mode of a square of seven 

 differences to the diagonal. This is shown in 3'1 to be 13 per cent. It was 

 therefore thought desirable to halve the differences and reapproximate. Body values 

 half-way between those of the smaller table were filled in by interpolation. 



Since the boundary formerly lay half-way between two sets of numbers, a set of 

 interpolated values now lies directly upon it, and consequently the surface conditions 

 resolve themselves into relations between the three outermost layers of numbers 

 instead of the two outermost as before. 



4'2'3'L The Method of Approximation. After some preliminary experiments, 

 the numbers just inside the boundary were corrected by taking Xm+i = X>~ a ~ 1 ^' 4 X'" 

 and after each approximation the numbers just outside the boundary were corrected 

 so as to keep 8x/% e( l ua l to ^xfiq, calculated analytically. The following values of 

 a were taken in turn: 10, 30, 50, 64, and again 2, 5, 20, 40, 40, 50, 60. These 

 numbers were chosen, as all such have been, because they gave a suitable curve of 

 reduction of amplitudes. 



4'2'3 - 2. An analytic integration of the boundary conditions was carried out, 

 using equations (6), (7) together with (11), (12) at the corners, and starting from 

 X = 0, Bx/S^ 0, behind the tail of the dam, as in the small table. The expressions 

 deduced for x and ^xfal are gi yen m a schedule. Values of x calculated from them 

 were set down in their places on the boundary, and the first difference across the 

 boundary was made equal to 3x/3<^. These numerical values may be seen in 

 Table XII. They have been carefully checked at a number of points by the theorem 

 about total changes given in 4'1'9, and have been found to be correct to O'Ol or 

 less. The exact values of x obtained from the boundary have been compared on 

 2 = with those found from the point-force and lake pressure in the bedrock 



a quite satisfactory agreement. 



