WITHDRAWN FROM THE ACTION OF GRAVITY. 425' 



does not dififer sensibly from a concave hemispliere, wliose diameter is conse- 

 quently equal to that of the tube. Let us recall, moreover, a part of the reason- 

 ing by which we arrive, in the theory of capillary action, at the law which 

 connects the height of the column I'aised with the diameter of the tube. Let us 

 suppose a pipe excessively slender proceeding from the lowest point of the hem- 

 ispheric surface in question, descending vertically to the lower orifice of the tube, 

 then bending horizontally, and finally rising again so as to terminate vertically 

 at a point of the plane surface of the liquid exterior to the tube. The pressures 

 corresponding to the two orifices of this little pipe will be, on the one part, P, 



2A 

 and, on the other, P — , if by <J be designated the diameter of the concave 



hemisphere, or, what amounts to the same thing, that of the tube. Now, the 

 two forces P mutually destroying one another, there remains only the force 



2A 



— , which, having a sign contrary to that of P, acts consequently fi-om be- 

 low upwards at the lower point of the concave hemisphere, and it is this which 

 sustains the weight of the molecular thread contained in the first branch of the 

 little pipe between the point just mentioned and a point situated at the height 



2A 

 of the exterior level. This premised, let us remark that the quantity -r- is the 



action which results from the curvature of the concave surface. The double of 



4A 



this quantity, or -;;-, will therefore express the pressure exerted on the enclosed 



air by a laminar sphero or hollow bubble of the diameter d, and formed of the 



same liquid. It thence results that this pressure constitutes a force capable of 



sustaining the liquid at a height double that to which it rises in the capillary 



tube, and that consequently it would form an equilibrium to the pressure of a 



column of the same liquid having that double height. Let us suppose, for the 



sake of precision, '5 equal to a millimetre, and designate by k the height at which 



the liquid stops in a tube of that diameter. We shall have this new result, that 



the pressure exerted on the enclosed air by a hollow bubble formed of a given 



liquid and having a diameter of 1™"^, would form an equilibrium to that exerted 



by a column of this liquid of a height equal to 2h. Now, the pressure exerted 



by a bubble being in inverse ratio to the diameter thereof, (§ 22,) it follows that 



the liquid column which would form an equilibrium to the pressure exerted by 



27i 

 a bubble of any diameter whatever, d, will have a height equal to ~. 



It would seem at first that this last expression ought to apply equally well 

 to liquids which sink in capillary tubes, h then designating this subsidence, the 

 tube still being supposed 1"^"^ in diameter ; but it is not altogether so, for that 

 would require, as is readily seen by the reasonings which precede, that the 

 surface which terminates the depressed column in the capillary tube should be 

 sensibly a convex hemisphere ; now we know that in the case of mercury this 

 surface is less curved ; according to the observations of M. Bede,* its height is 

 but about the half of the radius of the tiibe ; whence it follows that the valua- 

 tion of the pressure yielded by our formula would be too small in regard to such 

 liquids. It may be considered, however, as a first approximation. 



§ 24. Let us take as a measure of the pressure exerted by a bubble the height 

 of the column of water to which it would form an equilibrium. Then, if p desig- 

 nates the density of the liquid of which the bubble is formed, that of water being 

 1, the heights of the columns of water and of the liquid in question which would 



* Mcmoircs de VAcadcmie, tome xxv des Memoires couronnds et des Memoires des 6trangcrs. 



