98 



TRANSPORTATION OF DEBRIS BY RUNNING WATER. 



Now, in saying that one quantity varies as 

 a certain power of another, or in using such a 

 function as (28), the index of variation, or that 

 by which is indicated the comparative rate of 

 variation, is the exponent n; and the value of 

 this exponent, in terms of the variables, is found 

 not in (29) but in (31). The expression "y 

 varies as the nth power of x" is equivalent to 

 "the rate of variation of y, considered as a 

 power, is n times the rate of variation of x, 

 considered as a power." 



FIGURE 29. Logarithmic locus of the power function. 



An equivalent result would be attained if 

 attention were given to the quality of the 

 growth of x and y. Considering their growth 

 as occurring by additive increments, equation 

 (29) gives the ratio between their rates of in- 

 crease. Considering their growth as a matter 

 of multiplication by ratios, the additive incre- 

 ments are increments to logarithms, and equa- 

 tion (31) gives the ratio between rates of in- 

 crease. 



Equation (30), the logarithmic equivalent of 

 (28), is the equation of a straight line. Repre- 

 senting it by AB in figure 29, its inclination 



CO 



equals n, and its intersection with the axis 



of log y gives CO = log a. These familiar prop- 

 erties enhance the utility of function (28) by 

 enabling the investigator to discuss its con- 

 stants on logarithmic section paper. 



In many, probably a large majority, of the 

 physical problems to which the power function 

 is applied, it is found that the exponent, n, 

 does not have a constant value through the 

 entire range of observed values of x and y. 

 The locus of log y=/(log x} is then not a 

 straight line but a curve, which we may repre- 

 sent by AB in figure 30. At any point of the 

 curve C, its minute element, not distinguishable 



CE 

 from a straight line, has an inclination, jyp 



and which we may call n 2 . The value of n 2 

 varies from point to point of the curve, so that 

 if we try to express the relation of y to x in the 

 form of equation (28) we must regard the ex- 

 ponent as a variable. 



The element at C may also be thought of as 

 part of the tangent h'ne CD, of which the equa- 

 tion is 



log y = FO + ftj 

 whence 



On comparing this with (28), it is seen that 

 log-'FO corresponds to a. Let us replace it 

 by ,, giving 



yc,*"-.. (32) 



It is evident that for any other point of the 

 curve, as C', the tangent intersects the axis of 

 log y at a point different from F, and this cor- 

 responds to a different value of a t . In other 

 words, if we would express in an equation of 

 the type of (28) the same relation between two 

 variables that is expressed by the logarithmic 

 locus in figure 30, we must make the coefficient 

 as well as the exponent variable. 



which is homologous with n in (31) and (28) 



FIGURE 30. Locus of log y-/(log x), illustrating the nature of the index 

 of relative variation. 



The values of a, and n 2 are evidently func- 

 tions of the independent variable, x. 



It is possible to give to the relation shown 

 by the logarithmic locus an algebraic expression 

 which is identical in form with (28) and in 

 which the coefficient is a constant. That is to 

 say, it is possible to segregate the variability of 

 parameters in the exponent; but when that is 

 done the exponent no longer corresponds to the 

 expression "varies as the nth power of," and 



