SURFACES OF DISCONTINUITY 637 



to be permeable. The necessary amount of the fluids A and B 

 can be fed in from large reservoirs through narrow tubes let in 

 through the exterior envelop of the whole system, and the 

 liquid C can be passed out through a similar tube into a reservoir 

 of C in which the potentials and pressure can be adjusted; for 

 throughout this process the one variable is the pressure of the 

 fluid C in the gradually contracting lens. It is very necessary to 

 observe that for equilibrium at each stage of the process this 

 pressure increases with contraction of the lens, as can be readily 

 seen by considering the simple case of a spherical membrane 

 contracting with a constant external pressure on it and a con- 

 stant tension in it. This conceptual process may help the reader 

 to realize that the sentence near the bottom of page 263, 

 beginning: "It is not necessary that this should be physically 

 possible . . . ," is not an entirely arbitrary statement support- 

 ing a doubtful line of reasoning. Now let x stand for this 

 internal pressure which increases from a value p c which exists in 

 the fully formed lens and ends up at a larger value p c" when the 

 lens just disappears. During the process the values of the 

 surface areas between A and C, and between 5 and C will change, 

 and we will represent them as functions of x^ viz. Si{x) and S'iix), 

 respectively; the initial values of these functions are S>ac, Sbc 

 and the final values zero. The value of the part of the surface 

 which would lie between A and B extended into the lens, and 

 which decreases as the lens contracts, we will represent by 

 S3 (a:) ; its initial value is Sab and final value is zero. Similarly 

 Vi(x) and V2{x) will respectively represent the volumes between 

 the surface A-C and the surface A-B extended into the lens, 

 and between the surface B-C and the surface A-B so extended, 

 while V3{x) will represent their sum, the volume of the whole 

 lens at the stage when the internal pressure is x. The initial 

 values of Vi(x), V2(x) and ^3(2;) are Va, Vb and Vc respectively; 

 their final values are zero. Now consider the function of x, 

 f{x), defined by 



fix) = (TAcSiix) + (TBcSiix) — (Tab Si(x) 



+ Pa Vi{x) + Pb Viix) — xvi{x). 



