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



under the water. On the air surface the left-hand sides of these are zero. Here e is 

 the tangent of the angle which the surface, drawn to the right, makes with the 

 ; axis. 



We shall require the surface stresses in terms of the space-rates of x along the arc 

 and normal. The theory of stress functions is fully discussed by Mr. J. H. MICHELL 

 in ' Proc. London Math. Soc.,' Vol. XXI.; on p. 110 we find the required trans- 

 formation, which expressed in our notation reads as follows : 



, Tl -- 



as- 2 R dq Bs Bq R 9s 



These equations refer to a weightless solid, hence the suffix. To complete them 



we must add the stress zz = gpz, xx = = xz after transforming it to components 

 about the normal and arc. This gives 



4'17. The Size of the Dam. Suppose we have determined x for a dam of a 

 particular shape filled to a certain fraction of its height with water. Let us say 

 x , =/(:, 4 



We wish to find the stresses in a dam b times as big every way and containing 

 1) times the height of water. Try ^2 f( x /b, z/b). Then d\ 2 /dx 2 , S\ 2 /3z 2 , and 

 B^a/Sse 82 would all be l/b 2 times their former values at corresponding points of the 

 surface, but the constant terms in the surface conditions, due to the water pressure 

 and weight of masonry, are now b times their former values at corresponding points. 

 Consequently b x f(x/b, z/b) is the form of x appropriate to a dam b times the size of 

 the one for which f(x, y) was determined, and the stresses in the former will be 

 everywhere b times as great. For convenience in calculating, we will suppose that 

 the co-ordinate difference, 8x = Sz, is equal to the unit of length, unless otherwise 

 stated. The result can afterwards be applied to a dam of any size. 



4'1'8. Simple Transformation Concerning the Density p of the Masonry. If the 



reservoir be empty, and x* is the integral for density pi, then *- x* is the integral for 



density p, for it still satisfies the body equation V 4 ( x* ) = and also the surface 



\PI I 



equations (6) and (7), since both N and T vanish, so that the calculation of one 

 integral suffices for all sizes and densities, if it be multiplied by the proper constants. 

 Now when the reservoir is full we may take account of any density by means of 

 two independent integrals. Let us calculate x f r reservoir empty and density p l 

 and x f for reservoir full and density p 2 , then the proper stress-function for reservoir 



