424 THE FIGURES OF EQUILLBEIUM OF A LIQUID MASS 



cal figure of equilibrium in coucave. Now, these latter figures are evidently 

 realized by tlie surfaces of the laminar figures of equilibrium which face towards 

 the interior: the interior surface of a soap-bubble, for instance, is a spherical 

 surface in concave ; the interior surface of a laminar cylinder constitutes a cylin- 

 drical surface in concave, &;c. 



Pressure exerted hy a spJierical film on the air which it contains. — Application. 



§ 21. The exterior surface of a laminar sphere being convex in every direction, 

 the pressure which corresponds to it is greater than that of a plane surface, and 

 consequently (§ 10) the resultant of the pressures exerted in any point of the 

 bubble by the. two surfaces of the latter is directed towards the interior; whence 

 it results that the bubble presses on the air which it encloses. It is, indeed, 

 well known that, when a soap-bubble has been infiaLed, and while it is still at- 

 tached to the tube, if the other extremity of this last be left open, the bubble 

 gradiially collapses, expelling the air which it contained through the tube. We 

 see now what is the precise cause of this expulsion. 



§ 22. But we may go further, and determine according to what law it is that 

 the pressure, exerted by such a bubble on the confined air, depends on the diam- 

 eter of that bubble. We can compute, moreover, the exact value of the pres- 

 sure in question for a bubble having a given diameter and formed of a given 

 liquid. The pressure corresponding to a point of a laminar figure has (§ 9) for 



its expression A 8 r- -f — I . Now, in the case of the spherical figure, we 



have R = R' = the radius of the sphere. If, therefore, wo designate by d 



4A 

 the diameter of the bubble, the value of the pressure will simply become --=- , 



always, be it understood, neglecting the slight thickness of the film; whence it 

 follows that the intensity of the 2:)ressure exerted by a laminar spherical bubble 

 on the air which it confines is in inverse ratio to the dianaeter of that bubble. 



§ 23. This first result established, let us recur to the general expression of 

 the pressure corresponding to any point of a liquid surface, an expression which 



isP-1 1— + 77, j. For a surface of convex spherical curvature, if we des- 



2 \K W f 



ignate by d the diameter of the sphere to which this surface pertains, the above 



2A 

 expression becomes P + ^- , and for a spherical surfiice pf concave curvature 

 a 



2A . 



pertaining to a sphere of the same diameter, we shall have P -— . Thus, in 



the case of the convex surface, the total pressure is the sum of two forces acting 

 in the same direction — forces, of which one designated by P is the pressure 



2A . 



which a plane surface would exert, and the other represented by —j- is the ac- 

 tion which depends on the curvature. On the contrary, in the case of the concave 

 surface the total pressure is the difference between two forces acting in opposite 

 directions, and which are again, one the action P of a plane surface, and the 



2A 

 other —r- which depends on the curvature. Whence it is seen that the quan- 



4A 

 tity — , which represents the pressure exerted by a spherical film on the air it 



encloses, is equal to double the action which proceeds from the curvature of one 

 or the other surface of the film. 



Now, when a liquid rises in a capillary tube, and the diameter of this is suf- 

 ficiently small, we know that the surface which terminates the column raised 



