«L 'iTU'lKNCV OF TKKUKSTIAL JIUTATION lOi; THE I) K FLECTION OF STREAMS. [) 



P, — F, ovidi'iitl.v expiesst's tlif st'lcctivi- jiower due to lotiitioii, and /' — ./' siinilaily f^x])resses 

 the selective i)o\veidiie to ciirvatiue. ^^'llel^■ tlie curNanue has a couvexity to tlie liglit, tliese two 

 influences conspire, and their vesultant is deducible by aiUlition. Where the curvature has a left- 

 ward convexity, tlie influences arc opposed, and their resultant is dediicil)le by subtraction. [The 

 terminology here and throughout the renuiinder of the paper is adjusted to the northern hemis- 

 phere exclusively.] 



If we represent by K the joint scle(;tive jtower on curvatures of riglithand convexity and b\ 

 /. tlu' Joint selective i)ower (Ui curvatures of lelthand convexity, then we deduce, by simple com- 

 binations and transformations of equations (.S) and (4), 



^''^'•- + f- + - i^ii ••<iii A - 



L V, + i\ — - fjii sin A 



r, and r, may be the velocities of any two threads of current moving at ditferent rates, but 

 for purposes of convenience and simpliticatiou we now assume that they are symmetrically related 

 to the mean velocity r : and introducinj>' this relation in (5) we obtain 



R r -\- 1,11 sin A . . 



L r — pii sin A 



This ecjnation expresses the ratio between the selective luduences tending to determine the 

 maximum velocity toward the right and left banks respectively of a meandering stream. Since 

 these tendencies result in erosion, their ratio is a function of the tendency of a stream to erode its 

 right bank as compared with its tendency to erode the left. 



For the purpose of quantitative illustration, the Mississippi River will be considered. In its 

 lower course the shari)er bends have a radius of curvature, measured to the center of the channel, 

 of about S,Ono feet. These curves, together with all other channel features, are determined by 

 the water at its flood stage. It is therefore i)roper to consider in this connection the mean tlood 

 velocity. That was determined by Humphreys and Abbot to be, at Columbus, Ky., 8.4 feet per 

 .second.* The latitude of the locality is 31°. Giving these values to /j, v, and A, and substituting 

 for 11 its numerical value, .()(l(l07i;024, we obtain from (6). 



^1= 1.087. 



The .selective tendency toward the right bank is tiierefore nearly 1) per cent, greater than 

 toward the left. 



With the elements of another stream it is piobablc that a very different result would be obtained ; 

 but this single exam])le suffices to show that while the influence of rotation is small as compared 

 to that of curvature, it is still of the same order of magnitude, and may reasonably be expected to 

 modify the results of the more jxiwerful agent. In the present state of hydraulic science it is im- 

 jiossible to defDue the (piantitative relation between the tendency of swift threads of current toward 

 a bank and the consequent erosion ; but whatever that relation maybe, I conceive that rotation is 

 competent to produce appreciable results wherever those due to curvature are great. 



It will be observed that the ctticiency of rotation thus advocated is only in connection with, 

 and as an adjunct to, lateral wear by means of curvature. There are two general cases, including 

 a large share of all streams, to which the conclusion does not apply: (1) A stream which rapidly 

 conades the bottom of its channel docs not notably corrade its banks : and in such case the effect 

 of rotation should not be discoverable. [2) A stream engaged in the deiiosition of detritus, as on 

 a delta or an alluvial fan, shifts its chan'.ud from side to side by a process entirely distinct from the 

 one Just described. It builds up its bed until it is higher than the adjacent plain, and then trans- 

 fers its cunent bodily to aditfeient course, dotation has its share of influence in determining the 

 direction of this transfer; and it thereliy induces the stream to build its alluvial plain higher on 

 the right than on the left; but, the difference of level having been established, the stream has 



"Uniiipliii-.vs ;iii(l Alil'iit, Ivi'jimt in\ Mississippi River, p. .">9,^, 

 8. .Mis. m L' 



