VIl] 



OF PLATEAU'S BOURRELET 



299 



and all alike iu contact with air, the three surface tensions must 

 be equal ; and the three films must therefore, in all cases, meet 

 at an angle of 120°. But, unless the two bubbles be of precisely 

 equal size (and therefore of equal curvature) it is obvious that the 

 tangents to the spheres will not meet the plane of their circle 

 of contact at equal angles, and therefore that the partition-wall 

 must be a curved surface : it is only plane when it divides two 

 equal and symmetrical cells. It is also obvious, from the sym- 

 metry of the figure, that the centres of the spheres, the centre of 

 the partition, and the centres of the two spherical surfaces are 

 all on one and the same straight fine. 



Now the surfaces of the two bubbles exert a pressure inwards 

 which is inversely proportional to their radii : that is to say 

 ]) : J)' : : l/r' : 1/r ; and the partition wall must, for equilibrium, 

 exert a pressure (P) which is equal to the difference between these 



Fig. 104. 



two pressures, that is to say, P = l/R = l/r' — l/r = (r — r')/rr'. It 

 follows that the curvature of the partition wall must be just such 

 a curvature as is capable of exerting this pressure, that is to say, 

 R = rr'/{r — r'). The partition wall, then, is always a portion of 

 a spherical surface, whose radius is equal to the product, divided 

 by the difference, of the radii of the two vesicles. It follows at 

 once from this that if the two bubbles be equal, the radius of 

 curvature of the partition is infinitely great, that is to say the 

 partition is (as we have already seen) a plane surface. 



The geometrical construction by which we obtain the position 

 of the centres of the two spheres and also of the partition surface 

 is very simple, always provided that the surface tensions are 

 uniform throughout the system. If ^ be a point of contact 

 between the two spheres, and cp be a radius of one of them, then 

 make the angle cpm = 60°, and mark off on pm, pc' equal to the 



