Surface Forces in Fluids. 361 



results that the argument carries less conviction than it would 

 do if the relation between its hypotheses were more easily 

 apprehended. 



20. It is often a matter of interest to examine in what re- 

 lation one method of investigating a physical problem stands 

 to another ; and I shall endeavour in what follows to make 

 clear why the incorrect assumption of Laplace, that the den- 

 sity of the liquid is uniform, leads, from his method of con- 

 sidering the question, to the same equation to the surface of a 

 liquid as is obtained by Poisson, who takes the variation of 

 density into account. 



If we examine, as did Laplace, the difference of the action 

 on the upper portion of an elementary filament of unit section, 

 normal to a liquid surface at when the surface is curved, as 

 along AOB (fig. 9), from the action when the liquid is bounded 

 by the tangent-plane COD (which we will for convenience sup- 

 pose horizontal), we see that the difference is due to the excess 

 or defect of liquid above or below the level of CD ; and the 

 total differential action is due to the sum of actions between 

 all elements such as M of the canal and each of the elemen- 

 tary volumes such as EF, into which we may suppose the 

 differential matter divided. 



Let us imagine it to be divided in the following manner: — 



Let the sphere whose radius is the radius r of molecular 

 action be described about the element M, and let spheres of 

 radius r—dr, r — 2dr, &c. be also described about the same 

 point, so as to divide the differential matter under con- 

 sideration into spherical shells of equal elementary thickness 

 dr. Again, imagine that there pass through OP a series of 

 vertical planes, the inclination of each of which to the next is 

 the infinitesimal angle d6. The differential liquid comprised 

 between any two consecutive spherical surfaces is now par- 

 titioned off into infinitesimal solid elements ; and if we confine 

 our attention to all the elements contained between a single 

 pair of spherical surfaces, we see that the dimensions of each 

 in the direction of the radius of the spheres is the same, 

 viz. dr, and in the horizontal direction at right angles to this 

 radius is the same, viz. Og x d6 ; but in the remaining direc- 

 tion EF, perpendicular to the other two, the dimensions of 

 each element are not the same, but will depend in each case 

 on the curvature of the surface. Now, provided that the 

 curvature of the surface is always such that the radius of 

 molecular action is insensible in comparison with the least 

 radius of curvature, it follows that that portion of any normal 

 section of the surface made through the point 0, which falls 

 within the sphere of molecular action described about any 



Phil. Mag. S. 5. Vol. 18. No. 113. Oct. 1884. 2 B 



