V] OF LIQUID FILMS 217 



If instead of a cylinder, which is curved only in one direction, 

 we take a case where there are curvatures in two dimensions (as 

 for instance a sphere), then the effects of these riiust be simply 

 added to one another, and the resulting pressure j) is equal to 

 T/R + T/R' or p^TiljR+ l/R') *. 



And if in addition to the pressure p, which is due to surface 

 tension, we have to take into account other pressures, 7?', j)", etc., 

 which are due to gravity or other forces, then we may say that 

 the total 'pressure, P = p' + p" + T {l/R + l/R'). While in some 

 cases, for instance in speaking of the shape of a bird's egg, we 

 shall have to take account of these extraneous pressures, in the 

 present part of our subject we shall for the most part be able to 

 neglect them. 



Our equation is an equation of equilibrium. The resistance 

 to compression, — the pressure outwards, — of our fluid mass, is a 

 constant quantity (P) ; the pressure inwards, T {l/R + 1/-R'), is 

 also constant; and if (unlike the case of the mobile amoeba) the 

 surface be homogeneous, so that T is everywhere equal, it follows 

 that throughout the whole surface l/R + l/R' -= C (a constant). 



Now equilibrium is attained after the surface contraction has 

 done its utmost, that is to say when it has reduced the surface 

 to the smallest possible area ; and so we arrive, from the physical 

 side, at the conclusion that a surface such that l/R + l/R' = C, 

 in other words a surface which has the same mean curvature at 

 all points, is equivalent to a surface of minimal area : and to the 

 same conclusion we may also arrive through purely analytical 

 mathematics. It is obvious that the plane and the sphere are two 

 examples of such surfaces, for in both cases the radius of curvature 

 is everywhere constant, being equal to infinity in the case of the 

 plane, and to some definite magnitude in the case of the sphere. 



From the fact that we may extend a soap-film across a ring of 

 wire however fantastically the latter may be bent, we realise that 

 there is no limit to the number of surfaces of minimal area which 

 may be constructed or may be imagined ; and while some of these 

 are very complicated indeed, some, for instance a spiral helicoid 

 screw, are relatively very simple. But if we limit ourselves to 



* This simple but immensely important formula is due to Laplace {Mecanique 

 Celeste, Bk x. suppl. Theorie de Vaction capillaire, 1806). 



