ADJUSTMENT OF OBSERVATIONS. 



63 



cent, and crosses the line of zero capacity 

 before reaching the slope of 9 per cent. 



The general characteristics of stream traction 

 do not admit of a maximum in the relation 

 of capacity to slope. Capacity for traction is 

 clearly an increasing function of the stream's 

 velocity, and the velocity is clearly an increas- 

 ing function of the slope. There is reason also 

 to believe that capacity increases at an in- 

 creasing rate up to the slope corresponding to 

 infinite capacity. There are three forces con- 

 cerned in traction first, the force of the cur- 

 rent, of which the direction is parallel to the 

 slope; second, a component of gravity, when 

 gravity is resolved in directions parallel and 



normal to the slope; third, the resistance of 

 the bed, which is a function not only of the 

 others, but inversely of the slope. Within the 

 range of experimental slopes the component of 

 gravity is negligible in comparison with the 

 force of the current, and the influence of slope 

 on the resistance is relatively unimportant; 

 but as the angle of stability for loose material 

 is approached the resistance diminishes rapidly, 

 and at the slope of instability (65 to 70 per 

 cent for river sand) gravity is competent to 

 transport without the aid of current, and the 

 stream's capacity is infinite. All these factors 

 depend on slope, and as the increment to 

 capacity verges on infinity in approaching the 



-us 



+ 8 



(llaj 



/tea) 



.1 .2 .3 .4 .5 .6 .7 .8 .9 



Slope 



FIGURE 21. Extrapolated curves of C=/(S) for tentative equations of interpolation and for slopes less than 0.8 per cent. 



slope which limits variation, it is highly prob- 

 able that capacity grows continuously with 

 slope. 



This criterion suffices for the rejection not 

 only of the specific formulas (6a) and (lla), 

 but also of their types, (6) and (11). In for- 

 mula (6a) the occurrence of the maximum 

 value of C is determined by the negative coeffi- 

 cient of S 3 ; and it is true as a general fact that 

 equations of the class indicated by (12) yield 

 maxima whenever the coefficient of the highest 

 power of the independent variable is negative. 

 It is possible, or perhaps probable, that if each 

 series of laboratory values were to be formu- 

 lated under (7) or (8) the conditions for maxima 

 would be found to occur. On the whole, the 



extrapolations for higher slopes tend to restrict 

 choice to forms (9) and (10), with some reser- 

 vation as to forms (7) and (8). 



Figure 21 gives extrapolated curves for 

 slopes less than 0.8 per cent and represents the 

 same equations as figure 20, except that the 

 curve for (6a) is omitted. It will be observed 

 that it magnifies greatly the space between 

 and B in figure 20, the scale of slopes being 10 

 times and the scale of capacities 100 times as 

 large. The implications of the functions for 

 low slopes are specially important because 

 extrapolation from laboratory conditions to 

 those of natural streams will nearly always 

 involve the passage from higher to lower 

 slopes. 



