356 Mr. A. M. Worthington on the 



& 



plane, provided that at every point the least radius of curva- 

 ture is very great in comparison with the radius of molecular 

 action. 



Let a sphere whose radius is the radius of molecular action 

 be described about any ^.^L^e A at the surface of a liquid, 

 and let the figure (fig. 7) represent a section of the curved 

 surface containing the normal to the point A and the least 

 circle of curvature, and let B and C be the intersections of the 

 surface with the sphere ; PAQ the section of the plane tangent 

 at A whose intersections with the sphere are P and Q. Then, 

 if the radius AP is very small in comparison with the least 

 radius of curvature, the portion of the curve BAG is not to 

 be distinguished from the least circle of curvature, and BP or 

 CQ is very small in comparison with AP; and if a similar 

 normal section through A were made in any other plane for 

 which the radius of curvature was greater, the points corre- 

 sponding to B and C would be still nearer to the tangent- 

 plane. Also, if a smaller sphere were described about the 

 molecule A whose intersections in the plane of the paper 

 with the curved surface were b and c and with the tangent- 

 plane p and q respectively, then the ratio of the distance bp 

 to the radius Ap would be less than the ratio of BP to AP, 

 and the same would be true for any other normal section 

 through A. 



Now, in the investigation we have given, the terms F 1? F 2 , 

 F 3 , &c, have represented the action, on a given molecule, of 

 successive horizontal layers of molecules; and it is evident 

 that if each layer, instead of being flat, were bent so as to 

 be parallel to the slightly curved surface, the difference in the 

 action of each molecule of the layer, and therefore of the whole 

 layer on the given molecule, would be negligible in compa- 

 rison with the whole action of the layer. 



Again, the tension at any small depth below the surface 

 has been shown to be due to the variation, from the value in 

 the interior, of the repulsive action which it is necessary to 

 establish at that depth among evenly distributed molecules if 

 equilibrium is to be maintained, and the readjustment of the 

 molecular distances by which the equilibrium is secured in 

 nature has, as we have seen, the etfect of diminishing the 

 repulsive action still further, and of thus increasing the differ- 

 ence between its value at the given depth and the value in 

 the interior. 



Now the difference between the repulsive action of the first 

 and second layers was seen to be 



F 2 + F 3 + F 4 + ; 



