The Windings of Rivers 343 



-tream, following the windings, is 1-72 times greater than the 

 direct distance. In the Lahn we find almost the same factor. 

 Lower Danube, the Rhine, and the Ahr show a factor 

 approximating to 2. The Main, Mosel, Xeckar, and Thames 

 have lower factors. The mean of all the factors is 1-68. For 

 a certain number of the rivers the number of "bows" is given 

 with their average length. The size of the bows stands in 

 some relation to the volume of the river. What that relation 

 exactly is I am not able to state. To arrive at it will require a 

 careful study of the flood waters of the river in connection with 

 the form of its bed. It is the flood waters which form the bed. 

 When the river falls to low-water level we often see it cutting 

 out a secondary bed on a much smaller scale, which is oblite- 

 rated by the next following flood. 



It may be taken that the mean track of a stream traces the 

 lino of lowest level in the valley. Consequently, the ground 

 must rise on both sides of it. The cross-section of the valley 

 through the river resembles that through the middle of a watch 



5, rising at first very slightly on both sides of the stream, 

 then more rapidly as the confines of the valley are approached. 

 It is evident that water displaced to one side of the river will, 

 in returning to it, tend to pass to the other side, and to oscillate 

 about the lowest point. 



If the bed of a stream flowing through alluvial ground were 

 rectified so as to direct the water along a straight trough cut 

 in the material, it might preserve a straight course for a time, 

 but a stream following such a course is in a state of unstable 

 (juilihrium. The smallest accident or obstruction disturbs the 

 uniform rectilinear motion of the water, and tends to induce 

 "-.illation^ both longitudinal and transverse. These begin 

 immediately to cut into the banks, if they are yielding, and 

 take larger and larger dimensions until they reach a limit when 

 they have produced a course of the sinuosity which corresponds to 

 the laws of the harmonic motion of its waters. 



Noatt.mpt w.x made t<> arrive at t hose laws a priori. The 



od of investigation u-rd \\.i^ punlv empirical. Curves 



were traced according to ,11 kinds of harmonic specification 



until some \\crc- obtain, d \\hirh resembled the courses of actual 



