v] OF MINIMAL SURFACES 371 



essentially the same, save that the two halves of the one change 

 places in the other. 



That spheres, cylinders and unduloids are of the commonest 

 occurrence among the forms of small unicellular organisms or of 

 individual cells in the simpler aggregates, and that in the processes 

 of growth, reproduction and development transitions are frequent 

 from one of these forms to another, is obvious to the naturalist*, 

 and we shall deal presently with a few of these phenomena. 

 But before we go further in this enquiry we must consider, to 

 some small extent at least, the curvatures of the six different sur- 

 faces, so far as to determine what modification 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 impinging on the film, and also the 

 very important question of the limiting conditions which, from the 

 nature of the case, set bounds to the extension of certain of the 

 figures. 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 

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

 condition of our surface being one of minimal area. That is to say, 

 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, 

 IjR + IjR' = 0. The expression therefore vanishes, and our dy- 

 namical equation of equihbrium 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 sphere the radii are all equal, R= R'; they are also positive, 

 and T (l/R + l/R'), or 2T/R, is a- positive quantity, involving a 

 constant 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 TjR, to 



* They tend to reappear, no less obviously, in those precipitated structures which 

 simulate organic form in the experiments of Leduc, Herrera and Lillie. 



