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



preferred to the alternatives b and c, for with them the inconsistency was found to be 

 greater. It would be interesting to rework Table XV., taking, on the masonry- 

 water surface, numbers stepped out by finite differences exactly as was done for the 

 table with h = 1. It is not possible to adjust the boundary values of Table XV. so 



as to give by interpolation the infinitesimally correct values of ^' and -^- on the 



masonry-water boundary, for this would necessitate four independent adjustments in 

 the three values nearest to either corner. 



At the protuberant angles of 135 degrees the method () adopted for the re-entrant 



angles will not apply. For according to this method number (10) is not determined 

 by the boundary conditions (see Table XIII.). Neither is it determined by the body 

 equation, for it is right outside the boundary. But it cannot be omitted, for it enters 

 into the body equation at 1, 0. 



It has been derived from (10), as in the method (c). Fortunately a protuberant 

 angle of 135 degrees only occurs at one point, and even if the method adopted is not 

 quite correct, the stresses will only be affected in the immediate neighbourhood of the 

 angle. This point is more than half-way up the flank. 



4 '3. Conclusion. An attempt has been made to provide tables of the stress- 

 function, giving by their second differences stresses sufficiently accurate to be of use 

 to the practical designers of dams. The evidence that this has been done is : 

 (i.) The discussion of the boundary conditions in 4'1'11, 4'2'3'1, and 4'2'4 ; 

 (ii.) The discussion as to the completeness of the approximation in 4 '2 "2 "3 and 

 4 '2 - 3 '4 ; (iii.) The general agreement of the stresses derived from the three sizes of 

 co-ordinate difference in figs. 8 and 9. In this statement about the accuracy it is 

 assumed that the stress is taken from the table which has the smallest co-ordinate 

 difference. That is, from Table XV. having h = -j near the front toe ; from Table XII. 

 having h ^ elsewhere in the dam ; from Table X. having h = 1 deep down in 

 the bedrock. 



An additional confirmation is the strong resemblance of the curves in figs. 8 and 9 

 to those found by Messrs. WILSON and GORE* by stressing an india-rubber model 

 with a rounded angle at the front toe. 



The arithmetic has been carried out by a number of people, principally Messrs. 

 TILLEY, H. BOUCHARD, W. SHEPPARD, C. H. MASTERS, and G. EOBINSON, and I am 

 grateful to them for the care they have bestowed upon it. 



5. This work on the solution of physical problems by finite differences has been 

 carried on at intervals during three years. My thanks are due to Mr. G. A. SCHOTT 

 for convincing me of the desirability of discovering new methods for solving physical 

 problems; to Mr. A. BERRY, Mr. G. F. C. SEARLE, and Prof. KARL PEARSON for 

 their encouragement and advice in the early stages ; to Mr. A. CAMPBELL, 

 Mr. 3. C. M. GARNETT, and Mr. H. H. JEFFCOTT for references and the loan of books ; 



* ' Institution of Civil Engineers,' February, 1908. 



