v] OF LIQUID FILMS 367 



ward direction. In mathematical language, the pressm-e (p) varies 

 directly as the tension (T), and inversely as the radius of curvature 

 (R) : that is to say, p = T/R, per unit of surface. 



If instead of a cylinder, whose curvature lies only in one direction, 

 we take a case of curvature in two dimensions (as for. instance a 

 sphere), then the effects of these two curvatures must be added 

 together to give the resulting pressure p: which becomes equal to 

 T/R + TIR\ or 1 1 



R'^R' 



i.= H+-* 



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

 we have to take into account other pressures, p\ p", etc., due to 

 gravity or other forces, then we may say that the total pressure 



p=/+/'+2'(l+l) 



We may have to take account of the extraneous pressures in 

 some cases, as when we come to speak of the shape of a bird's egg ; 

 but in this first part of our subject we are able for the most part 

 to neglect them. 



Our equation is an equation of equihbrium. The resistance to 

 compression — the pressure outwards — of our fluid mass is a constant 

 quantity (P); the pressure inwards, T (IjR + l/R'), is also con- 

 stant; and if the surface (unlike that of the mobile amoeba) be 

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

 1/i? + l/R' = C (a constant), throughout the whole surface in question. 



Now equihbrium is reached after the surface-contraction has 

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

 the least possible area. So we arrive at the conclusion, from the 

 physical side, that a surface such that IjR + IjR' = C, in other 

 \<^ords a surface which has the same 7nean curvature at all points, 

 is equivalent to a surface of minimal area for the volume enclosed f ; 



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

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



t A surface may be "minimal" in respect of the area occupied, or of the volume 

 enclosed: the former being such as the surface which a soap-film forms when it 

 fills up a ring, whether plane or no. The geometers are apt to restrict the term 

 "minimal surface" to such as these, or, more generally, to all cases where the mean 

 curvature i» nil; the others, being only minimal with respect to the volume con- 

 tained, they call "surfaces of constant mean curvature." 



