_
/ [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.]
_
/| T23
/
3 /
<----------* 2
T31 1 \
\
_\| T12
The experimenters who have dealt with this question, C.G.M. Marangoni,
van der Mensbrugghe, Quincke, have all arrived at results inconsistent
with the reality of Neumann's triangle. Thus Marangoni says (_Pogg.
Annalen_, cxliii. p. 348, 1871):--"Die gemeinschaftliche Oberflache
zweier Flussigkeiten hat eine geringere Oberflachenspannung als die
Differenz der Oberflachenspannung der Flussigkeiten selbst (mit Ausnahme
des Quecksilbers)." Three pure bodies (of which one may be air) cannot
accordingly remain in contact. If a drop of oil stands in lenticular
form upon a surface of water, it is because the water-surface is already
contaminated with a greasy film.
On the theoretical side the question is open until we introduce some
limitation upon the generality of the functions. By far the simplest
supposition open to us is that the functions are the same in all cases,
the attractions differing merely by coefficients analogous to densities
in the theory of gravitation. This hypothesis was suggested by Laplace,
and may conveniently be named after him. It was also tacitly adopted by
Young, in connexion with the still more special hypothesis which Young
probably had in view, namely that the force in each case was constant
within a limited range, the same in all cases, and vanished outside that
range.