REVIEW OF CONTROLS OF CAPACITY. 



191 



a perfect formula competence would enter 

 twice. The formula adopted, however, ignores 

 the element of competence, chiefly because its 

 recognition, which would add a complication, 

 was not seen to be of advantage for the expres- 

 sion of the control of capacity in the more im- 

 portant regions outside the vicinity of compe- 

 tence. The accepted formula is 



The quantity p is that value of E which corre- 

 sponds to the maximum value of the maxi- 

 mum standing between the two zeros and 6 2 

 is a capacity constant. The function as a 

 whole qualifies capacity by means of a numer- 

 ical factor and may be combined with (91) by 

 multiplication of the factors: 



(109) 



The coefficient b, replacing 6 5 and b 2 , is a quan- 

 tity of the same unit with 6 6 (see p. 186), but 

 numerically independent. 



The function now added is of distinct type 

 from the others, for instead of advancing by a 

 continuous law from zero to infinity it first 

 rises to a finite maximum and then returns to 

 zero. The first three factors are harmonious; 

 the fourth discordant. At every stage in the 

 investigation the discussion of the laboratory 

 data has been hampered by this discordance. 

 In order to treat adequately the relation of ca- 

 pacity to either slope, discharge, or fineness, it 

 was necessary to isolate that relation by equaliz- 

 ing other conditions, and slope or discharge or 

 fineness could readily be equalized; but the 

 form-ratio factor was intractable. By means 

 of interpolation it was possible to assemble 

 varied data characterized by the same form 

 ratio, but that did not meet the difficulty. It 

 was necessary to take account of the relation 

 of the particular ratio to the optimum ratio, p; 

 and the value of ,0. varies with all other condi- 

 tions. 



The sensitiveness of capacity to form ratio, 

 as measured by the index of relative variation, 

 is less than its sensitiveness to other conditions. 

 The average of 48 values tabulated in Chapter 

 IV is 0.24, while similar averages for fineness, 

 discharge, and slope are three, five, and seven 



times as great. The distribution of sensitive- 

 ness, in relation to values of the independent 

 variables, is illustrated by figure 66, where four 

 curves are plotted, each representing a particu- 

 lar instance, selected as typical. The vertical 

 scale is the same for all ; and the ordinates rep- 

 resent values of the index of relative variation. 

 The horizontal scale is that of slope for the 

 curve SS, of discharge for the curve QQ, of 

 fineness for the curve FF, and of form ratio 

 for the curve BpR. The vertical cc represents 

 the competence constants and is an asymptote 

 to three of the curves. The horizontal line mm 

 gives the value of the exponent m correspond- 

 ing to the form-ratio index. For values of R 

 greater than p the index is negative, but its 

 curve is drawn above the zero line to represent 

 sensitiveness, which is not affected by sign. 



-q . 



FIGURE 66. Typical curves illustrating the distribution of the : 

 tiveness of capacity for traction to various controlling conditions. 

 Ordinates represent values of the index of relative variation; abscissas, 

 to four different scales, represent values of slope, discharge, linear 

 fineness, and form ratio. 



The two parameters of the form-ratio factor 

 have laws of variation similar to those of the 

 other parameters; each varies inversely with 

 values of all independent variables except its 



own, 



P =f(&,Q, h 



m=/, (&,Q,h 



(6D 



-62) 



It follows that each varies directly with each 

 of the other parameters (a, K, <p, n, o, and p). 



The relations of the parameters to form ratio 

 are less simple. In all cases the evidence from 

 the observational data is conflicting, and as the 

 several cases have come up for consideration 

 the trend of evidence has seemed now in one 

 direction and now in another. Impressions as 

 to those trends are recorded in the preceding 

 chapters, but when assembled they fail to indi- 

 cate any general principle. Recourse is there- 



