Theory of Surface Forces. 465 



the sum of tensions, is thus 



2(cr n — a n+ i) (cr n+1 — cr w+2 ). 



Hence, if c- w+1 be intermediate in magnitude between <r n and 

 <7 n+2 , the sum of tensions is increased by the abolition of the 

 stratum ; but, if cr n+ i be not intermediate, the sum is de- 

 creased. We see, then, that the removal of a stratum from 

 between neighbours where it is out of order and its intro- 

 duction between neighbours where it will be in order is 

 doubly favourable to the reduction of the sum of tensions; 

 and since by a succession of such steps we may arrive at the 

 order of magnitude throughout, w T e conclude that this is the 

 disposition of minimum tensions and energy. 



So far the results of Laplace's hypothesis are in marked 

 accordance with experiment ; but if we follow it out further, 

 discordances begin to manifest themselves. According to (47) 



y /T n =i l /T 1M +y/T m (48) 



a relation not verified by experiment. What is more, (47) 

 shows that according to the hypothesis T 12 is necessarily 

 positive ; so that, if the preceding argument be correct, no 

 such thing as mixture of two liquids could ever take place. 



But although this hypothesis is clearly too narrow for the 

 facts, it may be conveniently employed in illustration of the 

 general theory. In extension of (25) the potential at any 

 point may be written 



Y^a-n(f)dxdydz, (49) 



and the hydrostatical equation of equilibrium is 



dp = adV (50) 



By means of the potential we may prove, independently of 

 the idea of surface tension, that three fluids cannot rest in 

 contact. Along the surface of contact of any two fluids the 

 potential must be constant. Otherwise, there would be a 

 tendency to circulation round a circuit of which the principal 

 parts are close and parallel to the surface, but on opposite 



Ffr. 8. 



sides. For iri the limit the variation of potential will be 

 equal and opposite in the two parts of the circuit, and the 

 resulting forces at corresponding points, being proportional 

 also to the densities, will not balance. It is thus necessary to 

 equilibrium that tbere be no force at any point ; that is, that 

 the potential be constant along the whole interface. 



