﻿Intelligence and Miscellaneous Articles, 321 



size, and associated with massive bodies which he regards as copro- 

 lites. The author proposed to call this the "Lynton Bone -bed; " 

 and he thought that its discovery might throw some light on the 

 relative age of the whole series of rocks of North Devon. 



XLIII. Intelligence and Miscellaneous Articles. 



ON THE CURVE OF A RIVERAS BED. BY SAMUEL SHARPE, ESQ. 



To the Editors of the Philosophical Magazine and Journal. 



Gentlemen, 



MY friend Mr. Alfred Tylor, F.G.S., having by careful mea- 

 surement found that the beds of many rivers approach 

 more or less to the parabola, I was led to inquire whether under 

 any, and what, assumed conditions that curve would be the line 

 formed by water falling from a hard rock and flowing over a bed of 

 alluvial soil. 



In this case V, the velocity of the water on the bed of the river, 

 and R, the resistance of a grain of sand, vary at every different part 

 of the curve. But the conditions of the proposition require that the 

 difficulty with which a grain of sand is moved should be everywhere 

 the same when the river's bed has been cut into the form which it 

 afterwards keeps ; that is, 



— =a, a fixed quantity. 



XV 



Let C be the point from which 

 the curve starts, being the harder 

 soil, which the water does not 

 wear away, and A B a portion of 

 the curve =V. 



Let y be the ordinate, 

 x be the abscissa. 

 Then AB is the diagonal of a 

 parallelogram of which dx and dij are the two sides. 



The ease with which a grain is moved forward will be as -£ ; and 



dx 



its resistance R= b ® in which b is a fixed quantity. Then 



~=a becomes X~- =*, andV=a&— . 

 it o dx dy 



Then let the velocity of water falling down a perpendicular at B be 

 taken as ~cy 2 , in which c is a fixed quantity ; V its lessened velocity 



along the surface of the curve becomes =cy 2 -, supposing the water 



x 

 to reach B with the same velocity as if it had travelled along an in- 

 clined plane from C to B. 



