RELATION OF CAPACITY TO FORM RATIO. 



129 



will appear in another connection.) An in- 

 quiry as to the nature of the functions an 

 inquiry here unrecorded except as to its 

 result served to determine that the varia- 

 tions of and m^ as functions of R, affect 

 capacity in opposite senses, so that their in- 

 fluences are at least partly compensatory. It 

 appeared ' also that each influence is relatively 

 great when R is large and relatively small 

 when R is small, so that their laws of distribu- 

 tion include a compensatory factor. 



Despite the existence of this difficulty, 

 which is palliated rather than cured by features 

 of compensation, the function in (53) has been 

 adopted as the best practicable formula for 

 the relation of capacity to form ratio. The 

 logical defect in the combination of its two 

 factors may mean that its accuracy is inferior 

 to that of the factors; but as the factors are 

 confessedly only approximate, the defect is not 

 inconsistent with the possession of even superior 

 accuracy by the combination. In the treat- 

 ment of so intricate a subject by methods 

 which are dominantly empiric altogether ade- 

 quate formulation is not to be hoped for; but 

 the modicum of physical foundation afforded 

 to formula (53) is believed to give it advan- 

 tage over a purely mathematical expedient. 



A slight transformation gives it more con- 

 venient form. Moving R from the denomina- 

 tor to the numerator and making m l 1 = m, 

 we have 



C x (1 - a R) R m 



It is convenient also to change from a propor- 

 tion to an equation; introducing a coefficient, 6 2 , 



(7= & 2 (I a R) R m (54) 



<7=346 (I -2.18 R) 



.(55) 



The corresponding curve is shown in figure 41. 



This conforms to the physical conditions by 

 indicating two values of R for which capacity 

 is nil, and an intermediate value for which 

 capacity is at maximum. On referring to (54), 

 it is evident that <7=-0 when R = 0, and also 



when 1 n-7? = 0, or R = -. 



a 



In ascribing finite capacity to all small 

 values of R the formula is inaccurate, for when 

 the form ratio is gradually reduced the velocity 

 must always fall below competence before the 

 ratio reaches zero. The true function might be 

 represented by some such curve as the broken 

 line D in the figure. It has not seemed advis- 

 able to complicate the formula by a modifica- 

 tion which might remedy this defect. 



In the region of the larger values of R the 

 formula is subject to a qualification already 

 mentioned on page 125. The larger values are 

 associated with narrow channels, in which the 



As R is a ratio between lengths, and a (p. 125) 

 is also a numerical quantity without dimen- 

 sions, & 2 is of the unit of C. It is the value of 

 capacity when (1 a R) R m l. 



Let us now consider the properties of the 

 formula. For the sake of giving a visible illus- 

 tration, it has been applied to the example 

 already used in figure 38. As the equation has 

 three parameters, its constants require for their 

 determination three pairs of values of R and C. 

 The example furnishes four pairs, and these, by 

 approximate adjustment, give 



FIGURE 41. Plot of equation (55). Capacity, vertical; form ratio, hori- 

 zontal. 



influences of the side walls or banks are domi- 

 nant. So far as the laboratory data give indi- 

 cation, the formula is applicable to these values; 

 but the adjustment might be less satisfactory 

 with channel walls of a different character. 



The region of the maximum value of capac- 

 ity, which constitutes the chief field for the 

 application of the formula, is little affected by 

 the qualifications which have been mentioned. 



Differentiating equation (54) and equating 

 the first differential coefficient with zero, we 

 have 



Tf\ 



whence R = - -r, which is the condition giv- 

 am+1' 



ing C its maximum value. Designating this 

 value of R by p, we have 



m 



1 



nrm + 1' 



1 



a = ~ 



m 



p m + 1 



-(57) 



20921 No. 8614- 



