CAPILLAKY MOVEMEXT OF SOIL MOISTURE. 47 



EVALUATION OF EMPIRICAL CURVES, 



In order to determine "whether any mathematical relation could bo 

 found between the curves representing the movement of moisture in 

 the different soils, mathematical equations to fit these empirical 

 curves "were found for typical flumes. The curves representing the 

 movement of moisture in flumes at various slopes containing Eiver- 

 side heavy decomposed granite loam were evaluated to ascertain 

 whether the movement of moisture "was a function of the angle of the 

 slope. 



The problem of finding a mathematical equation to fit a given 

 curve is a tedious one. Since many soil physicists are perhaps un- 

 familiar with methods of procedure other than by the method of 

 least squares, which is so laborious as to limit its application, the 

 method which was used to derive these formulse is explained in 

 detail for two of these, one of which is a simple case and the other 

 much more complicated. The method used is that explained in 

 Engineering Mathematics, C. P. Steinmetz, Xew York. 1917, pages 

 209-274, to which reference is also made for an explanation of the 

 properties of different curves, 



In the following description, the number of days on which the 

 moisture position was observed is denoted, by x and the position of 

 advancing moisture measured in inches above the water surface is 

 denoted by y. The corresponding values of x and y were tabulated 

 and plotted as a curve. It is apparent that the curve in every 

 instance must pass through the origin, for when ay— 0, y=0. and 

 the nature of the problem also suggests that the curve be in the form 

 of a parabola. This was found to be true in the majority of cases, 

 but. as will be seen in the formula? given on a subsequent page, the 

 curve law in some instances changed within the range of the 

 observations. 



Curves which are represented by ?/=#a? n are parabolic or hyper- 

 bolic curves passing through the origin. When n is positive, the 

 curve is parabolic. When n is negative, the curve is hyperbolic. 



The logarithm of the equation y=ax n is log y=log a-\-n log x, 

 which is a straight-line formula. If the curve resulting from the 

 plotting of the logarithm of y against the logarithm of x is a straight 

 line, the curve representing the data is a parabola or hyperbola. 



The equation for the exponential curve is y=a^, which usually 

 occurs with negative exponent in the form y=az- nx : which gives log 

 y=log a—nx log s. Log y is a linear function of x and plotting log 

 y against x. or log x against y. gives a straight line. Thus plotting 

 log y and log x and x and y against each other permits the form 

 of curve to be recognized. If constant terms exist, the logarithmic 



