ON THE COHESION- OF FLUIDS. 



6G3 



Tins theorem may be very simply inferred 

 from llie former, by considering that, ac- 

 cording to the principle laid down in the 

 second section of this essay, the sum of the 

 thicknesses of the evanescent mcniscoid, in 

 any two planes passing tiirough the axis at 

 right angles to each other, is equal to the 

 sum of the thicknesses of the two menisci 

 formed by the largest and the smallest radii 

 of curvature ; consequently the sum of the 

 whole actions of these menisci must be twice 

 as great as the action of the meniscoid. 



" By means of this theorem, and of the laws of the equi- 

 Tibriuiii of fluids, we may determine the figure which 

 must be assumed by a gravitating fluid, inclosed in a vessel 

 of any given form. We obtain from these principles an 

 equation of partial differences of the second order, the in- 

 tegral of whidi cannot be found by any known method. If 

 the figure is such, as might be formed by the revolution-of 

 a curve round an axis, the equation is reduced to common 

 differences or fluxions, and its integral or fluent may be 

 found very near the truth, when the surface is very small. 

 I have shown in this manner, that, in very narrow tubes, 

 the surface of the fluid approaches the nearer to that of a 

 sphere, as the diameter of the tube is smaller. If these 

 segments are similar, in different tubes of the same sub- 

 stance, the radii of their surfaces will be" directly " pro- 

 portional to the diameters of the tubes. Now this similarity 

 of the spherical segments will easily appear, if we con- 

 sider that the distance, at which the action of the tube ceases 

 to be sensible, is imperceinible; so that if, by means of a 

 very powerful microscope, it were possible to mal<e it ap- 

 pear equal to the thousandth part of a metre, it is probable, 

 that the same magnifying power would augment the appa- 

 rent diameter of the tube to several metres. The surface 

 of the tube may therefore be considered as nearly plane, 

 within the limits of a circle equal in radius to the distance 

 at which its attraction becomes sensible ; consequently the 

 ftiiid within this distance, will be elevated or depressed 

 with respect to the surface of the tube, almost precisely in 

 the same manner as if it were perfecdy plane. Beyond this 

 distance, the fluid being subjected to no other sensible ac- 

 tion than that of gravitation, and that of its own attraction, 

 the surface will be very nearly that of a spherical begmeut, 

 the marginal parts of which, corresponding with those of 

 the surface of the fluid at the point which is the limit of the 

 sphere of the-stnsible activity of the tube, will be inclined 



very nearly in the same angle to its surface, whatever 

 its magnitude may be; hence it follows, that all these seg- 

 ments will be similar." 



The " near approach" of the surface of a 

 fluid in a very small tube to a portion of a 

 sphere, is sufficiently obvious from the funda- 

 mental principle, that thecurvatureis propor- 

 tional to the height above the general surface 

 of the fluid; for if the diameter of the tube be 

 small, this height will be so considerable, 

 that its variation at any part of the concave 

 or convex surface may be disregarded, and 

 the curvature may consequently be consi- 

 dered as uniform throughout the surface. It. 

 is only upon the sujiposition of a surface 

 nearly approaching to a spherical form, (hat 

 Mr. Laplace has endeavoured to determine 

 the " integral, very near the truth." He 

 has deduced from the expression, which 

 indicates the curvature of the surface, ano- 

 ther which is simpler, and which misht 

 easily have been inferred at once from the 

 uniform tension of the surface, as supportinu' 

 at each point the weiglitof the portion of the 

 fluid below it: he has then supposed this 

 weight to be the same as if the surface were 

 spherical, and Las deduced from this suppo- 

 sition an approximate expression, for the 

 elevation corresponding to a given anonlar 

 position of the surface only. This formula 

 is however still only apjilicable to those cases, 

 in wliich the suri'ace may be considered as 

 nearly spherical; and in these it is suj)crflu- 

 oiis. For example, if the surface of the 

 meicury in a barometer be depressed one 

 twentieth of an inch, as it actually is in n 

 tube somewliat less than a quarter of an inch 

 in diameter, Mr. Laplace's formula fails so- 

 completely, as to indicate a concavity in- 

 stead of a convexity ; for « being the reci- 

 procal of what I have called the appropriate 



