464 Lord Rayleigh on the 



so that by (42), (45), 



T u =(cr 1 -cr 2 ) 9 T (47) 



According to (47), the interfacial tension between any two 

 bodies is proportional to the square of the difference of their 

 densities. The densities <r u cr 2 , o- 3 being in descending order 

 of magnitude, we may write 



T 31 = (°"l ~ 0- 2 -f 0" 2 — C- 3 ) 2 T 



= T i2 + T 23 + 2(o- 1 — <r 2 ) (o- 2 — o- 3 )T ; 



so that T 31 necessarily exceeds the sum of the other two inter- 

 facial tensions. We are thus led to the important conclusion, 

 so far as I am aware hitherto unnoticed, that according to this 

 hypothesis Neumann's triangle is necessarily imaginary, that 

 one of three fluids will always spread upon the interface of 

 the other two. 



Another point of importance may be easily illustrated by 

 this theory, viz. the dependency of capillarity upon abrupt- 

 ness of transition. " The reason why the capillary force should 

 disappear when the transition between two liquids is sufficiently 

 gradual will now be evident. Suppose that the transition 

 from to a is made in two equal steps, the thickness of the 

 intermediate layer of density \<r being large compared to the 

 range of the molecular forces, but small in comparison with 

 the radius of curvature. At each step the difference of capil- 

 lary pressure is only one quarter of that due to the sudden 

 transition from to (7, and thus altogether half the effect is 

 lost by the interposition of the layer. If there were three 

 equal steps, the effect would be reduced to one third, and so 

 on. When the number of steps is infinite, the capillary 

 pressure disappears altogether."* 



According to Laplace's hypothesis the whole energy of any 

 number of contiguous strata of liquids is least when they are 

 arranged in order of density, so that this is the disposition 

 favoured by the attractive forces. The problem is to make 

 the sum of the interfacial tensions a minimum, each tension 

 being proportional to the square of the difference of densities 

 of the two contiguous liquids in question. If the order of 

 stratification differ from that of densities, we can show that 

 each step of approximation to this order lowers the sum of 

 tensions. To this end consider the effect of the abolition of 

 a stratum <r n +ij contiguous to a n and o- M+2 . Before the 

 change we have (<r n — a n+ i) 2 + (<r n+ i — o" w +2) 2 ? and afterwards 

 (a n — cr w+2 ) 2 . The second minus the first, or the increase in 



* a 



Laplace's Theory of Capillarity," Phil. Mag. October 1883, p. 315. 



