so that T31 necessarily exceeds the sum of the other two interfacial tensions. We are thus led to the important conclusion 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 abruptness 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 0 to σ is made in two equal steps, the thickness of the intermediate layer of density ½σ 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 capillary pressure is only one-quarter of that due to the sudden transition from 0 to σ, 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.” (“Laplace’s Theory of Capillarity,” Rayleigh, Phil. Mag., 1883, p. 315.)
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 σn+1, contiguous to σn and σn+2. Before the change we have (σn − σn+1)² + (σn+1 − σn+2)², and afterwards (σn − σn+2)². The second minus the first, or the increase in the sum of tensions, is thus
2(σn − σn+1) (σn+1 − σn+2).
Hence, if σn+1 be intermediate in magnitude between σn and σn+2, the sum of tensions is increased by the abolition of the stratum; but, if σn+1 be not intermediate, the sum is decreased. We see, then, that the removal of a stratum from between neighbours where it is out of order and its introduction 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, we 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 (52)
√T31 = √T12 + √T23, (53)
a relation not verified by experiment. What is more, (52) shows that according to the hypothesis T12 is necessarily positive; so that, if the preceding argument be correct, no such thing as mixture of two liquids could ever take place.
There are two apparent exceptions to Marangoni’s rule which call for a word of explanation. According to the rule, water, which has the lower surface-tension, should spread upon the surface of mercury; whereas the universal experience of the laboratory is that drops of water standing upon mercury retain their compact form without the least tendency to spread. To Quincke belongs the credit of dissipating the apparent exception. He found that mercury specially prepared behaves quite differently from ordinary mercury, and that a drop of water deposited thereon spreads over the entire surface. The ordinary behaviour is evidently the result of a film of grease, which adheres with great obstinacy.
The process described by Quincke is somewhat elaborate; but there is little difficulty in repeating the experiment if the mistake be avoided of using a free surface already contaminated, as almost inevitably happens when the mercury is poured from an ordinary bottle. The mercury should be drawn from underneath, for which purpose an arrangement similar to a chemical wash bottle is suitable, and it may be poured into watch-glasses, previously dipped into strong sulphuric acid, rinsed in distilled water, and dried over a Bunsen flame. When the glasses are cool, they may be charged with mercury, of which the first part is rejected. Operating in this way there is no difficulty in obtaining surfaces upon which a drop of water spreads, although from causes that cannot always be traced, a certain proportion of failures is met with. As might be expected, the grease which produces these effects is largely volatile. In many cases a very moderate preliminary warming of the watch-glasses makes all the difference in the behaviour of the drop.