462 Lord Rayleigh on the 



fluid which lie on the two sides of an ideal plane passing 

 through the interior, is per unit of area 2T l5 and the free 

 surface produced is two units in area. So for the second 

 fluid the corresponding work is 2T 2 . This having been 

 effected, let us now suppose that each of the units of area of 

 free surface of fluid (1) is allowed to approach normally a unit 

 area of (2) until contact is established. In this process work 

 is gained which we may denote by 4T' 12 , 2T' 12 for each pair. 

 On the whole, then, the work expended in producing two units 

 of interface is 2T, + 2T 2 —4T / 12 , and this, as we have seen, may 

 be equated to 2T ]2 . Hence 



T 12 =T 1 + T 2 -2T' 13 (42) 



If the two bodies are similar, 



1— J-2 — J- 12 > 



and T 12 = 0, as it should do. 



Laplace does not treat systematically the question of inter- 

 facial tension, but he gives incidentally in terms of his quantity 

 H a relation analogous to (42). 



If 2T' 12 > Ti + T 2 , T 12 would be negative, so that the inter- 

 face would of itself tend to increase. In this case the fluids 

 must mix. Conversely, if two fluids mix, it would seem that 

 T\ 2 must exceed the mean of Tj and T 2 ; otherwise work 

 would have to be expended to effect a close alternate stratifica- 

 tion of the two bodies, such as we may suppose to constitute a 

 first step in the process of mixture*. 



The value of T' 12 has already been calculated (7). We may 

 write 



T' 12 = 7T0-! a 2 \ e{z)dz = l7r < T 1 (T 2 \ z^(z)dz ; . (43) 

 Jo Jo 



and in general the functions 0, or <£, must be regarded as 

 capable of assuming different forms. Under these circum- 

 stances there is no limitation upon the values of the inter- 

 facial tensions for three fluids, which we may denote by T 12 , 

 T 23 , T 31 . If the three fluids can remain in contact with one 

 another, the sum of any two of the quantities must exceed the 

 third, and by Neumann's rule the directions of the interfaces 

 at the common edge must be parallel to the sides of a triangle, 

 taken proportional to T ]2 , T 23 , T 31 . If the above-mentioned 

 condition be not satisfied, the triangle is imaginary, and the 

 three fluids cannot rest in contact, the two weaker tensions, 

 even if acting in full concert, being incapable of balancing 

 the strongest. For instance, if T 3] > T 12 4- T 23 , the second 



* Dupre, he cit. p. 872. Thomson, Popular Lectures, p. 53. 



