RicHarpson— Lines of Flow of Water in Saturated Soils. 301 
Over a water-tight bottom with normal having direction- 
cosines /,, 7, %, we have 
a + My, - SF iy = BY 
Tf 2, m2, nz apply similarly to the upper surface, then the flow 
per unit area across it is 
op op Op 
l,—— + Mz + Ny — 
Ox Oy 0% 
: ‘ ehghiese: 
Now a vertical pipe of unit cross-section abuts on — units of 
9 
a 
sloping surface at the water-level. Therefore, if the surface is 
steady, the total flow in the said pipe is = of the above, that is, 
2 
ee eee 
2 OL U2 OY O08 seid 
Here W is the excess of rainfall over evaporation in volumes per 
area. If, on the other hand, the water-surface is moving, then we 
must take account of the capacity of the soil for moisture. Call 
it « volumes. of free water per unit volume of soil and water. 
Then, if s = @ be the equation for the free surface, 
oh Op 0p 0p 2 
W- ee (2 ae + mast + Ne =. (7) 
However, in what follows, I will only treat the case o 0. 
As usual, before attempting to solve a set of differential equations 
in detail, it is well to make a linear transformation of the 
variables. 
Let ¢, be determined as F (x, y, 2) by equations (4), (5), (6), 
and (7), and imagine that its values are recorded upon, or fixed 
into the substance of, an elastic medium of the size and shape of 
the piece of peat considered. Now, let this medium be stretched 
b times every way, while the coordinate axes remain graduated in 
centimetres as before. Then (4) is still satisfied. But the space- 
1 ; 
rate of ¢, is everywhere reduced to z of what it was. Now, 
suppose that the value of the function everywhere in the expanded 
