BELATION OF CAPACITY TO FOEM RATIO. 



131 



It is evident that 



increases with 



increase of R; therefore i 2 decreases with in- 

 crease of R. Also i 2 is positive when m > ^ -& 



aR 



approached but not reached, the positive 

 values of the index are all less than m. When 



aR = 1 , or R = -, the condition limiting traction 



<T 



for a narrow, deep stream, y = + GO 



and negative when m<-j ^. In passing 



,. l~. a tt and i.= oo. The negative values of the 



from positive to negative, i, passes through . , ,, , ~ - ., 



a ft ^ 1m index therefore range to infinity. 



zero when m = ^ -^ or when R=- ... , , =p The progressive changes of the index are 





(cf. equation 56). When R = 0, the limiting conveniently illustrated by the characters of 



a ft the curves in figure 42. Consider, for example, 



condition for a broad, shallow stream, 5^^5=0 the curve corre s p onding to m- 1.0. As this 



and i 2 = m. As this condition may only be curve is the logarithmic locus of C=f(R), its 



200 



Grade (G) 

 Q-.734 



S'Z.6-/, 



&6 



.Z A .6 



FIGUKE 43. Relation of capacity, C, to form ratio, R. The variation of the function Cbi(l<zR')R'n with elope. ?cale of C vertical; 



scale of R horizontal. 



inclination to the horizontal represents i 2 . At 

 the extreme left, corresponding to a very small 

 value of R (a shallow, wide stream), its inclina- 

 tion is a little less than 45 ; or the value of i 2 is 

 a little less than 1.0, the value of m. As the 

 curve ascends i 2 is positive. The inclination 

 then diminishes gradually and becomes zero at 

 the apex of the curve, which corresponds to 

 R p. As the curve begins its descent the 

 forward inclination corresponds to negative 

 values of i 2 , and these values increase numeri- 

 cally, becoming very large as the curve 

 approaches verticality. 



CONTROL OF CONSTANTS BY SLOPE. 



To illustrate the way in which the index of 

 sensitiveness and other constants defining the 

 relation of capacity to form ratio are influenced 



by the condition slope, two sets of computa- 

 tions were made. The first set used data per- 

 taining to debris of grade (C) and to a discharge 

 of 0.182 ft. 3 /sec.; the second, data for grade (G) 

 and 0.734 ft. 3 /sec. The observational data and 

 the computed constants appear numerically in 

 Table 27 and graphically in figure 43. The 

 dots in the figure represent points fixed by the 

 observations, and through these have been 

 drawn curves conforming to formulas (54) and 

 (58). Short vertical lines note the positions of 

 the maxima, where R = p. 



On referring to the table, it is seen that as the 

 slope increases the values of m, p, and a de- 

 crease, while the value of 6 2 increases. With 

 steeper slopes of channel, capacity is relatively 

 less sensitive to changes in the form of cross 

 section; with steeper slopes, the form of sec- 



