RicHarpson—Lines of Flow of Water in Saturated Soils. 303 
velocity-potential ¢,. ‘The lines of these two families must cut 
one another at right angles, forming rectangular chequers. 
In order that the body equation, V*#,=0, may be satisfied, 
length of chequer along normal to contour of ¢. 
breadth otf chequer along contour of @, 
the ratio: ; 
which we may call the “chequer-ratio,” must be constant along 
the normals to the contours of ¢,; and further, because each tube 
carries the same total flow, the chequer-ratio must be constant all 
over the field. The diagrams were sketched in pencil, rubbed out 
and repeatedly amended in the attempt to make the corners square 
and the chequer-ratio constant, with such success as you see. For 
a fuller account of the freehand graphic solution of V?V = 0, the 
reader is referred to a paper by the author published by the 
Physical Society of London, vol xxi. (also printed in Phil. Mag., 
Feb., 1908). 
The equations peculiar to this theory of drainage lead to the 
following conditions at the upper surface : 
By equation (12) The tubes of flow start from points which 
are equidistant horizontally—that is, each tube carries 
the same amount of rain-water. 
By equation (10) The equipotentials start from points which 
are equidistant vertically, as indicated by the faint 
lines on fig. 5. 
One finds, on making the drawings, that these conditions determine 
the shape of the surface very closely, in those figures in which the 
tubes of flow are somewhat horizontal. 
