December 5, 1907 J 



NATURE 



lOI 



almost the same factor. The Lower Danube, the Rhine, 

 and the Ahr show a factor approximating to 2. The 

 Main, Mosel, Necliar, and Thames have lower factors. 

 The mean of all the factors is i-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 rela- 

 tion 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 connec- 

 tion with the form of its bed. It is the flood waters 

 which form the bed. When the river falls to low-water 



Fig. 



level we often see it cutting out a secondary bed on a much 

 smaller scale, which is obliterated by the next following 

 flood. 



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

 the line 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 glass, 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 



specification until some were obtained which resembled the 

 courses of actual rivers. Fig. 2 shows one specimen out 

 of many which were exhibited at the meeting of the 

 British Association. 



It is assumed that the rhythmic motion set up in a 

 mass of water which is disturbed in its uniform rectilineal 

 motion will be reducible to two reciprocating motions, one 

 in the direction of the fall of the stream and the other at 

 right angles to it. When the gradient of the stream is 

 very steep and the nature of the bed homogeneous, as it 

 is in the case of water flowing down the front of a glacier, 

 the longitudinal oscillation is swamped 

 by tile powerful and continuous action 

 of gravity, which does not affect the 

 transverse component. In these cir- 

 cumstances we often meet with small 

 streams which describe an almost 

 perfect simple harmonic curve. 



In the ordinary stream of the 

 meandering type the gradient is very 

 small, in the case of the IVIississippi 

 from 2 inches to 4 inches per mile, 

 so that the longitudinal pulse can pro- 

 duce its full effect. When the two 

 oscillations are simple pendulum 

 motions and have the same period 

 they produce an ellipse, which, when 

 combined with the steady onward flow 

 due to gravity, produces sinuosities 

 unlilie those of actual streams. When 

 the period of the transverse oscilla- 

 tion is twice that of the longitudinal 

 one, their combination produces a 

 figure of eight (8). When a figure 

 of eight is combined with steady forward motion so 

 that both are travelled over in the direction of the 

 arrows in the figure, then it does delineate a curve 

 which may resemble the course of an actual stream. 

 This is illustrated in Fig. 2. In it the sinuous curve 

 falls into three parts, each consisting of a double 

 bow, corresponding to a complete excursion of the tracing 

 point round one of the figures of eight. The horizontal 

 line indicates the path of undisturbed flow of the stream 

 running from left to right in the direction of the arrows. 

 It is divided into seventy-two equal spaces, each of which 

 represents the distance which would be covered by the un- 

 disturbed stream in the interval of time in which the 

 circle which generates the transverse reciprocating motion 

 describes one twenty-fourth of a revolution, so that the 

 undisturbed stream passes over twenty-four spaces in the 

 time that the tracing point passes once round the figure 



Fig. 



is in a state of unstable equilibrium. The smallest 

 accident or obstruction disturbs the uniform rectilinear 

 motion of the water, and tends to induce oscillations, 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 

 h.-ivo produced a course of the sinuosity which corresponds 

 10 the laws of the harmonic motion of its waters. 



No attempt was made to arrive at these laws a priori. 

 The method of investigation used was purely empirical. 

 Curves were traced according to all kinds of harmonic 



NO. 1988, VOL. 77] 



of eight. The resultant path of the tracing point is the 

 sinuous curve, which cuts the horizontal line at 12 and 

 24 when the symmetrical 8 is used, and in 36, 48, or 

 60, 73 when one of the other two figures is used. It is 

 an essential condition that the tracing point shall go round 

 the 8 in the direction of the arrows, so that it shall be 

 moving in the same sense as the undisturbed water when 

 it traverses the outside parts of the figure which are 

 approximately parallel to the path of undisturbed flow. 

 In describing the sinuous line it is convenient to draw 

 the figure of eight on tracing paper. Then, when the 



