SURFACES OF DISCONTINUITY 539 



dividing surface from that which is termed the surface of tension 

 to one determined so as to make a specified surface concentra- 

 tion vanish. This is fully expounded in pages 233-237. In 

 the case of plane surfaces the term CiSci -f C25C2, which necessi- 

 tated the special choice of the surface of tension, disappears in 

 any case, and although es, rjs, Ti, r2, etc. will change in value 

 with a change in the location of the dividing surface, cr will not 

 change in value. To be sure, the proof given by Gibbs of this 

 statement is confined to plane surfaces, but it is easily seen to be 

 practically true even for surfaces of bubbles of not too great 

 curvature; for on using the equation p' — p" = a(ci -\- Ci) we 

 see that the increment of a caused by a change of amount X 

 in the position of the dividing surface, viz., X(ev" — ^v') 

 — t\{r]v" — riv') — mACti" ~ 7i') — etc., is not actually zero, but 

 equal to o-X(ci -f- Ci). As before, X, which is in all cases com- 

 parable with the thickness of the discontinuous region, is so 

 small that X(ci -f- C2) is an insignificant fraction, and so a is 

 altered by a negligible fraction of itself. A difficulty, however, 

 which might occur to an observant reader is the following. 

 Since a- is a definite function of the variables t, ni, ju2, etc., (for 

 so it has been stated), how comes it that da/ dm, da/dyLi, etc. 

 will alter with the location of the dividing surface? We have 

 just seen that cr does not alter, and certainly the variables t, m, M2, 

 etc. are in no way dependent on where we place the surface; 

 if (T is a definite function of t, m, H2, etc., so also are da/dni, 

 da/dni, etc. definite functions of the same variables, and appar- 

 ently they should no more change in value than a itself. The 

 solution of this difficulty requires the reader to guard against 

 confusing the value of a with the functional form of a. Actually, 

 if after the alteration a remained a function of the variables 

 t, Hi) M2, etc., the implied criticism would be valid; but a does 

 not do so. It must be borne in mind, as indicated by Gibbs on 

 page 235, that, with an alteration which makes Fi zero, a itself, 

 although not changed in value, has to be regarded as an entirely 

 different function, and moreover a function of the variables 

 t, 1J.2, jU3, etc., jui being excluded. The equation 



V'(i, Ml, M2, ...) = P"(t, Ml, M2, • . •) 



