APPLICATION TO NATUKAL STREAMS. 



239 



makes V vary as the square root of the hy- periments treat with confidence is from 0.5 to 



draulic radius, we have 3.0 per cent. Direct appli cation is limited to 



streams having slopes within that range. By 



-W-' = Z- 5 postulate, 



Then, since discharge is the product of width, 

 depth, and velocity, 



5_ T2.5 M141 



Q, 



F, 



Q,~w, 

 whence 



and 



Substituting this value of the fineness factor in 

 (113), and reducing, wo have 



.. .(115) 



The result indicates that 



whence 



- 



C, Q, 



^ 

 Q,, Q, 



C 



or, since -Q= U, the tractional duty of water, 



That is, for similar streams the tractional duty of 

 water is the same. 



As the exponents connecting capacity with 

 discharge, capacity with fineness, and mean 

 velocity with hydrauh'c radius are all averages 

 of low precision, the result is far from being so 

 secure as might be inferred from equation (115). 

 Its best support is really found in the plausi- 

 bility of its conclusion. Our experience with a 

 variety of physical laws makes it easy for us to 

 believe that with suitable parity of conditions 

 a unit of discharge will accomplish the same 

 work as part of a large stream that it will ac- 

 complish as part of a small stream ; and so the 

 conclusion is plausible. The fact that an at- 

 tempt to test the hypothesis of similar streams 

 by combining it with experimental data has led 

 to a plausible result is a fact favorable to the 

 hypothesis. 



Let us now assume the hypothetic law to be 

 a real law and draw such inferences as may be 

 warranted. The range of slopes which the ex- 



A/ "_ r ' ._ T 

 D, ~ V,,~ 



Substituting in equation (114), we have 



ef~\E7. 



whence 



0.5 

 1.0 

 2.0 

 3.0 



That is to say, for similar streams, the ratio ^- 5 



may be regarded as constant. This relation 

 affords a criterion for the discrimination of 

 those natural streams which are similar to the 

 laboratory streams, provided they are also 

 similar in slope and form ratio. The following 



limiting values for j^ for different slopes are 



all estimated on the assumption of a form ratio 

 of 0.05: 



Limiting values of " 



3,000,000-40,000,000 

 2,000,000-30,000,000 

 1,500,000-10.000,000 

 500,000- 4,000,000 



The form ratio 0.05 is considerably below the 

 average of the ratios developed in the labora- 

 tory, and it is also much above the average 

 for alluvial rivers at flood stage. Any allow- 

 ance which might be made for this discrepancy 

 would have the effect of increasing the estimate 

 of limiting values of the ratios of Q to .D 2 - 5 . 

 Subject to this qualification the ratios indicate 

 the types of natural streams which are "similar" 

 to the laboratory streams and to which various 

 laboratory results may be applied. The streams 

 are in general either small creeks or else rivers 

 transporting very coarse debris. As the slopes 

 are determined by flood discharges, such dis- 

 charges should be used in the classification. 



For the streams thus classified as similar to 

 laboratory streams the duty of water is of the 

 same order of magnitude, and so are the rates 

 of variation of duty with the several conditions 

 of slope, discharge, and fineness. The rates of 

 variation apply especially to comparisons of 

 one stream with another. For the estimation 

 of variation with discharge in the game stream 

 something should bo added to the laboratory 

 rate to allow for the varying assistance which 



