V] OF MINIMAL SURFACES 221 



is required, in each case, of the general equation which appHes 

 to them all. We shall find that with this question is closely 

 connected the question of the pressures exercised by, or im- 

 pinging on the film, and also the very important question of 

 the limitations which, from the nature of the case, exist to 

 prevent the extension of certain of the figures beyond certain 

 bounds. The whole subject is mathematical, and we shall only 

 deal with it in the most elementary way. 



We have seen that, in our general formula, the expression 

 1/R + 1/R' = C, a constant; and that this is, in all cases, the 

 condition of our surface being one of minimal area. In other 

 words, it is always true for one and all of the six surfaces which 

 we have to consider. But the constant C may have any value, 

 positive, negative, or nil. 



In the case of the plane, where R and R' are both infinite, it 

 is obvious that 1/R + 1/R' = 0. The expression therefore vanishes, 

 and our dynamical equation of equilibrium becomes P = p. In 

 short, we can only have a plane film, or we shall only find a plane 

 surface in our cell, when on either side thereof we have equal 

 pressures or no pressure at all. A simple case is the plane partition 

 between two equal and similar cells, as in a filament of spirogyra. 



In the case of the sphere, the radii are all equal, R ^ R' ; 

 they are also positive, and T {1/R + 1/R'), or 2T/R, is a positive 

 quantity, involving a positive pressure P, on the other side of the 

 equation. 



In the cylinder, one radius of curvature has the finite and 

 positive value R ; but the other is infinite. Our formula becomes 

 T/R, to which corresponds a positive pressure P, supplied by the 

 surface-tension as in the case of the sphere, but evidently of just 

 half the magnitude developed in the latter case for a given value 

 of the radius R. 



The catenoid has the remarkable property that its curvature in 

 one direction is precisely equal and opposite to its curvature in 

 the other, this ^property holding good for all points of the surface. 

 That is to say, R = — R' ; and the expression becomes 



{1/R+ 1/R') ^ {1/R- 1/R) = 0; 

 in other words, the surface, as in the case of the plane, has no 



