78 PROCEEDINGS OF THE GEOLOGICAL SOCIETY. 



make the relative quantity supplied to cd greater than that supplied 

 to Ac and Bd in the proportion as the distance between cd is less 

 than the distances between Ac and Bd ; consequently, in fig. 5, the 

 water-wear between cd would be slightly greater (aided also by the 

 tendency of the water to converge at P) than between Ac, Bd, and 

 the cavity, or incipient pipe, would increase in magnitude more 

 rapidly in the direction of cd, than of Ac, Bd, or in other words 

 would deepen more rapidly than it widened. Further, as the 

 dimensions of the pipe increased, so would the disproportion be- 

 tween Aa and Ac, and between hB and Bd constantly increase ; and 

 as only the same relative quantity of water would pass over the 

 surfaces Ac, Bd, whatever their dimensions might become, its effect 

 would be one gradually diminishing, and consequently the lateral 

 growth of the tube would tend to become less from day to day ; 

 whilst, as the proportion of cd with regard to ah would continue 

 with little variation whatever the size of the pipe, the relative 

 quantity of water passing on the surface cd would remain constant 

 or nearly so, and the downward growth of the pipe would continue 

 unimpaired. Thus the dimensions of Aa, ah, hB to Ac, cd, and dB 

 in fig. are nearly equal ; but in fig. 6, Ac and Bh are five times 

 greater than Aa and hB, whereas cd retains nearly the same propor- 

 tion to ah that it did in fig. 5, or the erosive effect of the water passing 

 Aa, hB being spread over an area five times larger would have be- 

 come five times less ; on the contrary that passing ah, acting upon an 

 area of the same dimensions as at first, would possess its original erosive 

 power, one now five times greater than that acting upon the sides. 



Consequently the action on the sides of the pipe would tend 

 to become, in process of time, comparatively insignificant, whilst 

 that on the base of the pipe retained its original force ; therefore 

 the water-channel would pass from a slight hollow to a funnel- 

 shaped cavity, and then into an indefinitely prolonged cylinder 

 with a pointed end. The points ah are of course arbitrary*, but 

 wherever we place them with regard to AB, cd will always exhibit 

 the same constant character with regard to ah ; and Ac, Bd the 

 reverse with regard to Aa, hB. Practically, also, a portion of the 

 water falling on the sides Ac, Bd would tend, from the ob- 

 struction presented by the partially impermeable lining of the 

 pipe, to be thrown off from those surfaces and pass through the 

 central area cd. 



With respect to the adjacent chalk surface, it may be observed 

 that, whilst the pipes were forming, the water would at the same 

 time be scoring and corroding that surface with conducting furrows 

 and channels converging in these absorbent cavities, and producing 

 much of the furrowed and worn surface apparent on the chalk when 

 freed from its superincumbent sands or gravel. 



* Nor is this intended to give more than a rough, but, I believe, at the same 

 time a correct general sketch of the process. For a more exact calculation, which 

 could easily be made, it would be necessary to take the relation of the superficial 

 area at the top of the pipe to that of the sides, and not the merely linear measure- 

 ment adopted above. 



