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 2T1 + 2T2 - 4T'12, and
this, as we have seen, may be equated to 2T12. Hence

T12 = T1 + T2 - 2T'12 (47)

If the two bodies are similar,

T1 = T2 = T'12;

and T12 = 0, as it should do.

Laplace does not treat systematically the question of interfacial
tension, but he gives incidentally in terms of his quantity H a relation
analogous to (47).

If 2T'12 > T1 + T2, T12 would be negative, so that the interface would
of itself tend to increase. In this case the fluids must mix.
Conversely, if two fluids mix, it would seem that T'12 must exceed the
mean of T1 and T2; otherwise work would have to be _expended_ to effect
a close alternate stratification of the two bodies, such as we may
suppose to constitute a first step in the process of mixture (Dupre,
_Theorie mecanique de la chaleur_, p. 372; Kelvin, _Popular Lectures_,
p. 53).

The value of T'12 has already been calculated (32). We may write

_
/ [oo]
T'12 = [pi][sigma]1[sigma]2 | [theta](z)dz =
_/0
_
1 / [oo]
= -- [pi][sigma]1[sigma]2 | z^4 [phi](z)dz; (48)
8 _/0

and in general the functions [theta], or [phi], must be regarded as
capable of assuming different forms. Under these circumstances there is
no limitation upon the values of the interfacial tensions for three
fluids, which we may denote by T12, T23, T31. 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 T12, T23, T31. 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 T31 > T12 + T23, the second fluid spreads itself
indefinitely upon the interface of the first and third fluids.

[Illustration: FIG. 5.]