RELATION OF CAPACITY TO SLOPE. 



109 



S, Q, and F. Parallel complexities would also 

 arise if attempt were made to formulate the 

 relations by means of such an expression as 



From equation (10) (p. 96), 



The sensitiveness of capacity to slope appears 

 to be a function of the conditions jointly 

 rather than severally. 



The development of complexity within com- 

 plexity suggests that the actual nature of the 

 relation is too involved for disentanglement by 

 empiric methods, but that conclusion does not 

 necessarily follow. Just as a highly complex 

 mathematical expression may be the exact 

 equivalent of a fairly simple expression of a 

 different type, so a physical law may defy for- 

 mulation when approached in a certain way 

 yet yield readily when the best method of 

 approach has been discovered. 



FORMULATION WITH CONSTANT COEFFICIENT. 



For the relation of capacity to slope the 

 formula equivalent to (34) is 



<7=<VSV ..(40) 



in which the constant coefficient c t is the value 

 of capacity when 8=1. 



(41) 



Also as (10) and (40) give equivalent expres- 

 sions for 0, we may equate their second mem- 

 bers: 



Cj^'-^OS-a)" 



Substituting the value of c t from (41), 



whence 

 and 



S-a\ n 



S 1 ' 



_ log(S- g ) -log (! 

 - n ~ logS 



.(42) 



By means of this formula the values of j\ in 

 Table 16 were computed. These values have 

 been subjected to systematic comparison with 

 the associated conditions (S, Q, w, etc.) in the 

 same manner as were the values of the ex- 

 ponent \ ; but the results of the comparison 

 need not be given in detail because they are 

 parallel to those for ij. They are in fact iden- 

 tical so far as verbal statement is concerned 

 and differ only in quantitative ways. It is 

 true in the main, and almost without excep- 

 tion, that the variations of j\ are less rapid 

 than those of i,; and their average range is 

 about half as great. 



TABLE 16. Values ofj lt in 0=^8 h, the coefficient c, being a constant. 



Grade..! 



