OBSERVATIONS ON DETERMINATIONS OF DROUGHT-INTENSITY. 73 



the velocity of the liquid varies directly as its density, the rate of 

 fall in pressure in the direction of flow, and the square of the linear 

 dimensions of the interstices of a stratum, and inversely as the 

 viscosity of the liquid. That is, if U denote the mean interstitial 

 velocity, K a constant, p the density and rj the viscosity of the 

 liquids, R any homologous linear measure of the size of the inter- 

 stices, and P the difference of pressure between two points L apart, 

 between which the velocity is uniform — so that dP/dL is the rate 

 of fall of pressure — then the expression 



U=K-t~ R* (1). 



rj dL 



defines all the laws of flow. Apart from experiment this formula 

 may be deduced rationally: i.e., by abstract mechanics. The value 

 of the constant iTcan also be obtained mathematically — i.e., with- 

 out recourse to experiment — when the form of the interstices is 

 given, and the particular dimension denoted by R is indicated. 

 For water p may be taken as sensibly unity: and, other things 

 being equal, the change of rate of flow with temperature will be 

 as follows : — 



Temperature Celsius 0°. 10°. 20°. 30°. 40° C. 



Rate of flow (/) 1-00 137 1-77 2-22 2-72. 



This covers the entire range of conditions naturally occurring. 

 Hence, if we put / for the rate of fall in pressure — i.e., for the 

 'hydraulic gradient' — the above equation may be written 



U=f'K'IR* (2) 



in which R measures the coarseness of the soil, and /' = p\y\ ; for 

 water very nearly f as above. 



8. Relation of rate of rainfall to saturation. — In order to clearly 

 apprehend the manner in which the saturation of a soil is depen- 

 dent upon the mutual relation of rate of rainfall and permeability, 

 consider the case of a uniform unsaturated soil of indefinite depth, 

 with a uniform surface slope, and subject to rainfall of varying 

 degrees of intensity. For every rate of fall — dz/dt say, since fall 

 is conveniently measured by the height (z) to which any level 

 closed surface is covered in some definite time (t) — less than what 



