

BY FINITE DIFFERENCES OF PHYSICAL PROBLEMS, ETC. 329 



full and density p will be Xf~~X* + PX<> f r > on accoun t of linearity, the solutions of 



\2 -i ^ 



when added together satisfy - - ^- -^- n 2 gpz = Ni + N a + N 3 . Here N t is the 



cs -t\ oo 



normal stress due to the water pressure of a full reservoir, and N 2 and N 3 are the 

 same when the reservoir is empty, that is zero. The above is the method of PEARSON 

 and POLLARD (p. 28) translated into stress-function symbolism. It shows us how to 

 find the stresses for a full reservoir sustained by masonry of any density by 

 calculating in detail two cases only. 



In this paper the case of full reservoir and density 2'25 is the only one treated. 



4'1'9. Integration of Surface Equations. We see from equations (G) and (7) that 

 if the shape of the boundary and the stresses upon it are given, then starting at a 

 point s a , and assuming initial values of ^, ?x/^> anc ^ ^X/^ s > we can mi( l :l double 

 row of values of x all round the boundary by straightforward integration. The 

 initial values of x, 3^/3f/, and 3x/3s are not significant, for they depend only on the 

 arbitrary function Atc + Bz + C, which may be added to any distribution of x without 

 affecting the stresses. The only outstanding uncertainty is at a sharp corner where 

 l/R becomes infinite. Prof. PKARSON has shown that at a sharp re-entrant corner 

 the stress may become infinite (' History of Elasticity,' vol. ii., 1711). In view of 

 this it appeared possible that the boundary strip would be indeterminate at the 

 corners, and the following investigation was made to settle the question. As it is a 

 question of infinity, the finite forces N ds, T ds, and gpzdsdq in the immediate 

 neighbourhood of the corner may be neglected, leaving simply 



3s 2 " 11 dq ' 3s 8(7 R 3s ' 



Now let us suppose that the trace of the boundary on the y plane is a circle at the 

 corner, with the intention of making the radius indefinitely small after integration, 

 on integrating equations (8) and (9) with R constant it is found, after some work, 

 that 



+ H (10) 



is the complete integral, C, D, and H being arbitrary constants. 



Now suppose that s/R, the angle turned through, increased from to . Then it 

 can be shown from (10) that 



cos + sin , = cos - sin . (11), (12). 



/o \d/9 \3/ \S/ \3s/c 



VOL. CCX. A. 2 U 



