102 



gravity then is to pnll these arcs of circles out into cycloidal-like curves 

 crowding near I) C. As a matter of fact the curve drawn from x = 5, 

 z ^ 10 is nearly a cycloid. Those in the upper left-hand corner being too 

 low and long and those in the lower right-hand corner too short and high 

 for cycloids. 



The lines of pressure are liyperbola-like curves drawn for pressures, 

 1, 2, 3, 4, etc., all the curves beginning and ending in the boundary. 



It is easy to see that we may take a similar area a b to the right of 

 A B C D and leaving an open face similar to A D and an impervious bot- 

 tom and water at zero pressure along the top. We should then have these 

 two areas one on each side of B C with the liquid flowing in opposite 

 directions. The li(iuid in each area flows directly doAvn B C and so the 

 motion will not be interrupted if B C be removed. That is. the method 

 of images is applicable horizontally. If. however, a similar area to A B 

 C D be taken just Vtelow C D wo can not say that the method of images 

 as usually applied holds true. We may regard A D in the upper area as 

 an absorbing slit and A D in the lower area as a similar slit and the 

 position C D between them as a mirror the corresponding iiarts of A D 

 in the upper and lower slits are not found at equal distances above and 

 below C D. They are found drawn down by gravity so that the 

 method of images must be modihed for vertical distrilmtions. By 



9 9 



integrating u with respect to z between the limits b and -- . b; jq ^ and 



~ . b, etc., tlie amount of flowage from each of the ten equal divisions of 

 A D may be calculated. And in a similar way the amount of liquid going 

 in at each of the ten equal divisions of A B is obtained by integrating 



,. . ,9 9 ,8 



w with respect to x between the limits a and jQ-a; j q • ^ and jq • a> 



etc. The equations for the flowage and the amount absorbed are then: 



, n TT (b — z) 

 n = 00 cosii 7i n, 



f- rd udz- ^^^ ^" - . cos^ 



t_ j udz _ _ . iiTTb 2a I 



^ '^ 11 — 1 u^ cosh J 



n zr: 1 11 ' cosh ^^ 



c 



^ n-(b — z) 

 n ^ CO cosh Tz .. _ -t-1 d Id 



a- ro 



J ^ ^' ,^-l u^ cosh '1^ ''^ J c J 



2a 



