RELATION OF CAPACITY TO SLOPE. 



99 



the expression thus lacks the essential quality 

 of (28). To illustrate, let us assume that the 

 position of the point F is fixed, so as to give a 

 constant value to a^ The values of n } then 

 correspond to the inclinations of lines FC, FC' , 

 etc., drawn to points on the curve; but these 

 lines are not tangents and their inclinations are 

 not those of the corresponding elements of the 

 curve. 



In order to satisfy the condition, from equa- 

 tion (28), that y = a when x=l and logz = 0, 

 the fixed point must be at the intersection, G, 

 of the curve with the axis of log y. The lines 

 connecting it with other points of the curve are 

 therefore chords. 



In the discussions of our laboratory data use 

 will be made of both these variants of the power 

 function; and they will be distinguished, from 

 one another as well as from (28), by the follow- 

 ing notation. In 



y=vx i 



(33) 



the coefficient and exponent are both variable 

 and are functions of x. The symbol v is chosen 

 for the coefficient to signalize its variability. 

 The exponent, i, denotes the instantaneous ratio 

 of the variation of y to the variation of x, when 

 those variations are viewed as ratios. It is tho 

 first differential coefficient of log y with respect 

 to log x, and it will be spoken of as the index of 

 relative variation. 

 In 



y = cxJ (34) 



the coefficient is constant and the exponent 

 variable. 



Whenever, in the investigation of the natural 

 law connecting two variables, pairs of simulta- 

 neous values of the variables are known by ob- 

 servation, it is possible to plot a curve represent- 

 ing empirically log y =/ (log 2) such a curve as 

 A CGB in figure 30. The directions of the ele- 

 ments of that curve, or the values of i, are es- 

 sentially facts of observation. They depend 

 exclusively on the phenomena and are inde- 

 pendent of the units in which observational 

 data are expressed. It is different with the 

 values of j, for those depend on the position of 

 the point, G, in which the curve intersects the 

 axis of log y, and therefore on the position of 

 the axis. The position of the axis corresponds 



to log x = or x = 1 and is thus dependent on 

 the magnitude of the unit by which the inde- 

 pendent variable is measured. 



In other chapters of this report much atten- 

 tion is given to the values of i, and the discus- 

 sion of the variations of such values is used as 

 a mode of treating empirically the relations 

 between the various factors of the general 

 problem of traction. 



THE SYNTHETIC INDEX. 



Recurring to figure 30, let us give attention to 

 a restricted portion of the curve, for example, 

 the part between C and C 1 . The value of i cor- 

 responding to the point C is the inclination of 

 the line OF; the value of i corresponding to C' 

 is the inclination of C' F'. Between the two are 

 a continuous series of other values. The incli- 

 nation of the chord connecting C and C', con- 

 considered as a ratio or exponent, is interme- 

 diate between the extreme values of i. If the 

 sequence of values follows a definite law, the 

 value given by the chord equals some sort of a 

 mean derived from the others ; and, in any case, 

 it is in a sense representative of the group. It 

 may be called a synthetic index of relative varia- 

 tion between the indicated limits. 



If the coordinates of C be log x' and log y', 

 and the coordinates of C' be log x" and log y", 

 then, representing the synthetic index by 7, 



.(35) 



/= logy" -log y' 

 log x" log x' 



As the direction of the chord depends on the 

 positions of G and G' upon the curve, so the 

 value of 7 depends on the limits between which 

 it is computed. As the direction of the chord 

 gives no information concerning the direction 

 of any part of the curve, so the value of 7 can 

 not be used to determine any particular value 

 of i. It is used in the following pages for the 

 comparison of different functions for which the 

 data span approximately the same range of 

 conditions. 



APPLICATION TO THE SIGMA FUNCTION. 



Let us now represent the relation of capacity 

 to slope by an equation of form (33), 



G= Vl S''.. ..(36) 



