THE AGE OF THE WYE. 
193 
From these figures, which leave out the constant 110 yards, it 
may be perceived that the breadth very nearly varies directly as 
the square of the length. If the lengths, therefore, be taken 
along a horizontal line, and the corresponding breadths perpendi- 
cular to that line, the ends of the co-ordinates will be in a 
parabolic curve, of which the apex will be at the zero point ; 
therefore, from a well-known property of this curve, the average 
“length” from the beginning until now will be just two-thirds of 
what it is at the present time, or J053 yards. 
A parabolic curve will not, however, truly represent the facts 
in an extreme case. The conditions will be more correctly 
represented by tangential circles of a radius equal to the average 
radius of curvature of the “ bends ” ; but, as the co-ordinates 
due to these tangential circles do not depart materially from the 
parabolic curve in the small arcs required for our question, the 
difference it would make in our reckoning of the average length 
of the benJs does not claim further notice. If we call the 
average radius of curvature R, the lengths L, and the corres- 
ponding breadths W, the formula for the co-ordinates due to the 
tangential circles is 16 R W — 4 W^ = L^, which is the equation 
to an ellipse whose axes are 8 R and 4 R, and the average 
length, in our case, becomes very slightly more than two-thirds 
of the present length. 
Let us consider next the process by which the sandstone 
rock is worn away. It has been shown tha.t the cause of wear 
is mainly atmospheric ; that, though mechanical attrition may do 
a little, the chief action of the stream is to wash away the 
particles of sand already loosened by the air. We may conclude, 
therefore, that the rate of wear will be greatly more rapid from 
the 110 yards of cliff fully exposed than from the 1580 yards 
covered by the alluvium, and thus in a great degree protected 
from the direct action of the air. 
