362 Mr. A. M. Worthington on the 



point M of the normal filament OP, cannot be distinguished 

 from the circle of curvature of the section in question. 



But if the radius of molecular action be insensible in com- 

 parison with the least radius of curvature of the liquid at the 

 point 0, then the length EG will be insensible in comparison 

 with the radius Mg, and the action of the element Eg on the 

 element M may be regarded as taking place along any line 

 joining M to any point of EF; and this will be the case whether 

 the density of the liquid be constant throughout EF or whether 

 it vary, provided only that the variation is insensible for 

 depths which are themselves insensible in comparison with the 

 radius of molecular action. 



Now, if the radius of curvature of the section containing the 

 element EF were different, viz. / (but still so great that the 

 radius of molecular action is insensible in comparison with it), 

 so that length EF of the element in question were to become 

 EF 7 , then, since the change of curvature evidently produces 

 no change in any other dimension of the element, the ratio 

 EF to EF' is the ratio in which the volume of the element is 

 altered, and it is easily shown, from the geometry of the 

 circle, that 



EF : EF' : : - : \ 



r r 



and the same is true for the corresponding element in the 

 section at right angles to the plane containing the line OP 

 and the element EF, or, using a corresponding notation, 



E*:E*'::i:*; 



whence, adding these two ratios, 



EF + E0:EF' + E(//::i+- : 1+A,. 

 ■ r p r p 



Hence, if we consider the attractions as proportional to the 

 volumes, we get that the attractions of any pair of cor- 

 responding elements of the differential liquid, in planes con- 

 taining the filament OP and at right angles to each other, is 

 proportional to the sum of the reciprocals of the radii of the 

 circles of curvature in those two planes. 



Now the sum of the reciprocals of the radii of the circles 

 of curvature at any point in any two planes at right angles to 

 each other is, by geometry, always equal to the sum of the 



reciprocals of the principal radii of curvature, =— + — ,; and 



consequently the total integral action of the differential liquid 

 above or below the plane surface, being made up of pairs of 



