APPLICATION TO NATURAL STREAMS. 



227 



equals AHxt&n IIAII^ and is proportional to 

 a diameter of the particle. On the assumption 

 that the diameters of the particle are equal it 

 follows that the volume of the wedge is pro- 

 portional to that of the particle, or to D 3 . The 

 distance of the center of gravity of the wedge 

 from the plane A^A^E^B^, being a linear di- 

 mension of the wedge, is proportional to D, and 

 the mean distance of transfer, which is double 

 that distance, is also proportional to D. The 

 quantity of motion normal to the shearing 

 planes, occasioned by the particle, is measured 

 by the wedge of water times the mean distance 

 of transfer, and, the mass of the wedge being 

 proportional to its volume, the quantity of 

 motion is proportional to D 3 xD = D*. The 

 quantity of motion occasioned by the entire 

 suspended load, all its particles being assumed 

 to have the same size, is proportional to ND 4 , 

 where N is the number of particles ; and if the 

 mass of the total load remain unchanged while 

 its degree of comminution is varied, it is evident 

 that N varies inversely with D 3 . Therefore the 

 quantity of internal motion occasioned by a 

 particular mass of suspended matter of uniform 

 grain is proportional (since D 4 /D 3 = D) to the 

 diameter of its particles. The fundamental 

 assumption of the analysis is that this motion 

 measures a resistance to the freedom of the 

 water which is coordinate with viscosity and 

 which may for practical purposes be considered 

 as an addition to the resistance arising from 

 the viscosity of the water. 



That portion of the viscosity factor which 

 depends on the molecular influence of the 

 particle outside its boundary is still less sus- 

 ceptible of quantitative estimate but may yet 

 be discussed with reference to the diameter of 

 the suspended particle. In ignorance of the 

 exact nature of the influence and also of the 

 law by which it diminishes with distance out- 

 ward from the boundary, I assume arbitrarily 

 that at all distances from the particle less than 

 I the freedom of water molecules is restricted, 

 and that the amount of restriction is measured 

 by the volume of the space in which the restric- 

 tion is experienced. The imperfection of this 

 assumption will of course affect any deduction 



from it. That volume is J TT (D + 2l) 3 - ~ r. D 3 . 



For the entire suspended load, assumeil to 

 consist of N equal particles, the total volume 



is N times as great. For a load of invariable 

 weight but variable comminution, N<XJ^, and 

 the total volume is proportional to 



D 3 



or, introducing a constant, It, and reducing, we 

 have 



Volume = 



Evidently the influence of this factor varies 

 inversely with D. When D is very large as 

 compared to 2l, it approaches zero; when D is 

 small as compared to 2l, it is relatively very 

 great. It is most sensitive to the control by D 

 when the particles are very small. 



The two divisions of the viscosity factor vary 

 in their influence on velocity with the comminu- 

 tion of the load but in opposite ways, the 

 influence of the first being greater as the 

 particles are larger and that of the second as 

 they are smaller. The laws of variation are 

 such that their combination exhibits a mini- 

 mum that is, for some particular size of 

 particle the influence on velocity is less than 

 for particles either larger or smaller. 



Another mode of treating the viscosity 

 factor assumes that, so far as the viscosity 

 effect is concerned, the molecular influence is 

 equivalent to an enlargement of each particle 

 to the extent of I on all sides. Then, reasoning 

 as before with reference to interference with 

 shearing, we obtain 



Resistance oc 



(D + 21Y 

 D 3 



This expression is not only simple but has the 

 advantage of giving definite indication of the 

 position of the minimum. The resistance is 

 least when Z? = 6Z. 



We may now bring together the qualitative 

 results of analysis, and write 



V m + 



This may be read: The mean velocity (F^,) of 

 a stream carrying a load of suspended debris 

 of diameter D equals the mean velocity (F m ) 

 of the same stream when without load, plus a 



